How price controls reallocate surplus

WEBVTT
Kind: captions
Language: en

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- [Instructor] What
we're going to talk about

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in this video is the
effect of price controls

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on changing how the surplus,

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the total surplus is reallocated between

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consumers and producers.

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And we already touch on
this in other videos.

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The video on rent control.

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The video on minimum wages.

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And so this is to make sure that we are

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taking away some of the big ideas.

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So, right over here I have my classic

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demand and supply curves.

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Of course, the rental market.

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At a high price

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the quantity demanded is low.

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The quantity that people would be willing

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to supply is quite high.

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And at a low price, the quantity

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that people would be willing to supply

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is low, while the quantity demanded

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would be quite high.

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And we've seen from many videos so far

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in our journey through economics

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that we have our equilibrium price

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and our equilibrium quantity

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where these two curves,
or lines intersect.

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So, I'll call this Q sub zero

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and this is price sub zero.

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But let's say for whatever reason

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city officials in this city

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where this rental market

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that this chart describes
the rental market for

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they decide that P sub zero is too high

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that their owners are
complaining that rent

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are too high in the market.

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And so the city decides
to put in a price control.

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And in this case, they try
to implement a price ceiling.

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So they say, look, the price of rent

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per square foot per month

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cannot go above this level

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right over here.

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So this is the price

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ceiling.

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Price ceiling, right over there.

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Now, what's going to happen here?

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Well, if this is the price ceiling,

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then right over here, this is

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the total amount of square footage

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the quantity of

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I guess, square footage

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that is being willing
that people are willing

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to supply that.

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The landlords, or building owners

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are willing to supply.

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But at this price,

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you have a much higher quantity

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that is being demanded.

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This is right over here
is the quantity demanded.

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And when the quantity demanded at a price

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is higher than the quantity supplied

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well, then you have a shortage.

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Put this right over here.

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Is describing

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a

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shortage.

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And we talk about that in other videos.

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But let's think about what's happening

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to the total surplus.

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So when we let the market just get to an

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equilibrium price and quantity

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the total surplus, actually let me just

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draw separately the consumer

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and the producer surplus.

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So this was the consumer surplus.

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Right over here.

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Before the government intervention.

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And then, this is the producer surplus.

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And we've talked about
this is other videos.

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But now what happens

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when we have this price control?

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Well, this is the quantity supply.

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Now, all of a sudden,

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the total surplus shrinks.

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The total surplus is now being depicted

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by this white trapezoid.

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And also, think about
how things have shifted.

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So one thing that you see clearly,

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what is the producer surplus now?

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The producer surplus is

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only this little yellow

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triangle, or this little.

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I don't know my colors.

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This little blue triangle at the bottom.

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So you see very clearly

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that all producers suffer here.

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All producers

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all producers suffer.

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Now you might say, well, of course

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this is a rent control.

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But surely, the consumers
will benefit here.

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Well, it isn't the case,
it is the case that

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some consumers, the ones that are able to

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get into a unit, they might benefit.

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So this right over here, some

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demanders, or you can say

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some consumers

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consumers

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benefit.

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But, not all of the consumers benefit.

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In fact, you have a shortage now.

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Before you had more people who were able

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to get housing.

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Now these folks are not
going to get housing.

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And, as in all of
economics you should take

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a grain of salt

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in any type of

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oversimplified model like this.

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Or simplified, this is actually

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a very useful model

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for thinking about certain things.

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Because, even these
consumers that are benefiting

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according to this model

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for these consumers it
looks like their surplus

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has grown, or if you're
this kind of marginal

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renter right over here,

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let me draw it over here.

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If you are the marginal renter

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that was before, getting

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this much benefit.

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Now you're able to get
that plus all of that.

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But think about the things that this model

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is not capturing.

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What's the incentive for the landlord now?

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Would they want to invest
in the building as much?

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Would they

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would they maybe let the building

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kind of suffer a little bit?

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And there's also particular quirks

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for how rent control is implemented

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that might also change behavior.

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So always keep in mind

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what's not captured by the model.

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But in broad brush terms

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you put in a price control, in this case,

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you put in a price ceiling

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you're going to create a shortage.

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All the producers are going to suffer.

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Some of the consumers

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benefit, according to this model.

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But not all of them.

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Because not all of them
are now going to be able

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to get a place to rent.

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Now let's move over to another market.

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Let's say the corn market.

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And let's say

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once again

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we have our equilibrium price

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and our equilibrium quantity.

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So price of equilibrium.

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And this is our

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quantity sub equilibrium.

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But let's say in this
situation the government

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let's say the farmer, the corn farmers

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are able to organize.

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And they're able to lobby
the government and say,

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hey, we really suffer

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when there's low corn prices.

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So where we want to
institute a price control

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we want you to institute a price control.

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We want a price war.

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So the government says, okay corn farmers

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you seem to be pretty serious about it

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so we are going to
institute a price floor.

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So the price cannot go

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below this level.

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So this is a price floor.

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Well, what's going to happen now?

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Well let's think about the quantity

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demanded, and the quantity supplied.

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So that, right over there,

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is the quantity demanded.

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You see where the price
intersects the demand curve.

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And this right over here
is a quantity supply

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where we intersect in the supply curve.

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Quantity supply.

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So in this situation

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your demand is less

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the quantity demanded

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is less than the quantity supplied.

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So in this case,

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you have a surplus.

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The farmers would want to produce more

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than people would want

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at that price.

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And also think about what happens to

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this total surplus.

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And in particular, the consumer
and the producer surpluses.

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So in the old world

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this was

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the consumer surplus.

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And,

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this right over here is

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the producer surplus.

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In the new world, the
total surplus shrinks

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the sum of the two.

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We're now talking about

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this, the area of this trapezoid.

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Right over here, this sideways trapezoid.

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And you see that all consumers suffer

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because now the consumers surplus

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has been shrunk right over

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the consumer surpluses right over here.

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There's consumers who are
now not even consuming corn.

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And even the ones that are consuming corn

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before, let's say this marginal consumer

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right over here was getting
all of this benefit.

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Now, they are getting a smaller benefit.

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So we could say, all

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all consumers

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suffer.

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Now what about the producers?

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Well, the producers who are able to

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sell their corn

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definitely get a benefit.

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So if you were this marginal producer

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right over here

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your surplus

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right over here

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would have been like that.

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But now you get even
a higher price for it.

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And so, one way to think about it

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is, some producers,

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some producers for sure, benefit

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according to this model.

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So let me write that.

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So, some

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producers

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producers benefit.

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But it's important to realize, once again,

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that not all of the producers benefit.

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Because once again, we have a surplus.

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If it's implemented this way

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well, you might have a lot of farmers

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who aren't even able to

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sell their corn

00:08:15.176 --> 00:08:16.253
at that price.

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So it's an interesting
thing to think about.

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You should always take
models with a grain of salt.

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But it is a pretty interesting framework

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where governments often will try to do

00:08:23.596 --> 00:08:25.433
some type of knee jerk solution

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to try to make something look good,

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or feel good to their constituents.

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But the end effect is that people might

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suffer more than they expect.

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They might cause a shortage when you put

00:08:34.764 --> 00:08:36.395
a price ceiling.

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Or, it might cause a surplus

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when you have a price floor


Proof of expected value of geometric random variable

WEBVTT
Kind: captions
Language: en

00:00:00.827 --> 00:00:02.510
- [Instructor] So right
here we have a classic

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geometric random variable.

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We're defining it as the
number of independent trials

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we need to get a success where
the probability of success

00:00:11.623 --> 00:00:14.040
for each trial is lowercase p

00:00:14.932 --> 00:00:17.433
and we have seen this before
when we introduced ourselves

00:00:17.433 --> 00:00:19.227
to geometric random variables.

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Now, the goal of this
video is to think about

00:00:21.099 --> 00:00:23.433
well what is the expected value

00:00:23.433 --> 00:00:27.388
of a geometric random variable like this

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and I'll tell you the answer,

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in future videos we
will apply this formula,

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but in this video we're actually going to

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prove it to ourselves mathematically.

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But the expected value of
a geometric random variable

00:00:38.384 --> 00:00:41.652
is gonna be one over the
probability of success

00:00:41.652 --> 00:00:43.941
on any given trial.

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So now let's prove it to ourselves.

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So the expected value
of any random variable

00:00:51.076 --> 00:00:54.252
is just going to be the
probability weighted outcomes

00:00:54.252 --> 00:00:55.448
that you could have.

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So you could say it is the probability...

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The probability that our
random variable is equal to one

00:01:02.094 --> 00:01:04.198
times one plus the probability

00:01:04.198 --> 00:01:07.560
that our random variable
is equal to two times two

00:01:07.560 --> 00:01:09.900
plus and you get the general idea.

00:01:09.900 --> 00:01:11.387
It goes on and on and on

00:01:11.387 --> 00:01:14.494
and a geometric random
variable it can only take on

00:01:14.494 --> 00:01:17.482
values one, two, three,
four, so forth and so on.

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It will not take on the value zero

00:01:19.198 --> 00:01:20.791
because you cannot have a success

00:01:20.791 --> 00:01:23.347
if you have not had a trial yet.

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But what is this going to be equal to?

00:01:25.583 --> 00:01:27.613
Well, this is going to be equal to,

00:01:27.613 --> 00:01:29.658
what's the probability
that we have a success

00:01:29.658 --> 00:01:31.180
on our first trial?

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And actually let me
just write it over here.

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So this is going to be P.

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What is this going to be?

00:01:36.788 --> 00:01:39.726
What is the probability
that we don't have a success

00:01:39.726 --> 00:01:42.335
on our first trial, but we
have one on our second trial?

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Well, this is going to be one minus p,

00:01:45.751 --> 00:01:48.167
that's the first trial where
we don't have a success,

00:01:48.167 --> 00:01:50.834
times a success on the second trial

00:01:50.834 --> 00:01:53.092
and actually let me do
a few more terms here.

00:01:53.092 --> 00:01:56.726
So let me erase this a little
bit, do a few more terms.

00:01:56.726 --> 00:02:00.773
So this is going to be the
probability that X equals two.

00:02:00.773 --> 00:02:05.692
Sorry, the probability that
X equals three times three

00:02:05.692 --> 00:02:07.493
and we're gonna keep
going on and on and on.

00:02:07.493 --> 00:02:09.227
Well, what's this going to be?

00:02:09.227 --> 00:02:11.449
Well, the probability that X equals three

00:02:11.449 --> 00:02:15.616
is we're gonna have to get
two unsuccessful trials

00:02:17.104 --> 00:02:19.003
and so the probability of
two unsuccessful trials

00:02:19.003 --> 00:02:21.072
is one minus P squared

00:02:21.072 --> 00:02:24.740
and then one successful
trial just like that.

00:02:24.740 --> 00:02:26.593
So you get the general idea.

00:02:26.593 --> 00:02:29.160
If I wanted to rewrite this
and I'm just gonna rewrite it

00:02:29.160 --> 00:02:31.033
to make it a little bit simpler.

00:02:31.033 --> 00:02:34.974
So the expected, at least for
the purposes of this proof,

00:02:34.974 --> 00:02:38.780
so the expected value of X is equal to,

00:02:38.780 --> 00:02:42.697
I'll write this as 1p
plus 2p times one minus p

00:02:46.535 --> 00:02:49.285
plus 3p times one minus p squared

00:02:51.179 --> 00:02:53.073
and we're gonna keep
going on and on and on

00:02:53.073 --> 00:02:54.345
forever like that.

00:02:54.345 --> 00:02:56.388
So how do we figure out this sum?

00:02:56.388 --> 00:02:58.872
And now I'm going to do a little bit of

00:02:58.872 --> 00:03:02.993
mathematical trickery or
gymnastics, but it's all valid

00:03:02.993 --> 00:03:04.394
and if any of ya'll have seen the proof

00:03:04.394 --> 00:03:07.402
of taking an infinite geometric series,

00:03:07.402 --> 00:03:10.047
then we're gonna do a
very similar technique.

00:03:10.047 --> 00:03:12.026
What I'm gonna do here
is I'm gonna think about

00:03:12.026 --> 00:03:15.041
well what is one minus p
times this expected value?

00:03:15.041 --> 00:03:16.381
So let's do that.

00:03:16.381 --> 00:03:20.548
So if I say one minus p times
the expected value of X,

00:03:21.492 --> 00:03:23.490
what is that going to be equal to?

00:03:23.490 --> 00:03:25.573
Well, I would multiply
every one of these terms

00:03:25.573 --> 00:03:26.591
times one minus p.

00:03:26.591 --> 00:03:28.942
So one p times one minus p

00:03:28.942 --> 00:03:31.442
would be 1p times one minus p.

00:03:33.837 --> 00:03:35.416
You would get that right over there.

00:03:35.416 --> 00:03:37.175
What about 2p times one minus p?

00:03:37.175 --> 00:03:38.764
What would that be equal to?

00:03:38.764 --> 00:03:42.084
Well, that would be 2p times one minus p

00:03:42.084 --> 00:03:44.814
and now we're gonna multiply
it by one minus p again.

00:03:44.814 --> 00:03:47.412
So you're gonna get one minus p squared

00:03:47.412 --> 00:03:49.763
and so I think you see where this is going

00:03:49.763 --> 00:03:51.762
and we're just gonna
keep adding and adding

00:03:51.762 --> 00:03:53.059
and adding from there.

00:03:53.059 --> 00:03:55.457
Now we're gonna do something
really fun and interesting,

00:03:55.457 --> 00:03:58.169
at least from a
mathematical point of view.

00:03:58.169 --> 00:03:59.618
If this is equal to that,

00:03:59.618 --> 00:04:01.504
if the left-hand side is
equal to the right-hand side,

00:04:01.504 --> 00:04:04.392
let's just subtract this
value from both sides.

00:04:04.392 --> 00:04:06.771
So on the left-hand side I would have

00:04:06.771 --> 00:04:09.740
the expected value of X, that's that,

00:04:09.740 --> 00:04:12.157
minus this, minus one minus p

00:04:13.519 --> 00:04:16.019
times the expected value of X.

00:04:17.613 --> 00:04:19.951
So I'm just subtracting
this from that side,

00:04:19.951 --> 00:04:22.301
but let me subtract this from that side.

00:04:22.301 --> 00:04:24.189
Well, I could subtract
this expression from that,

00:04:24.189 --> 00:04:25.022
but this is equivalent,

00:04:25.022 --> 00:04:27.952
so I'm just gonna subtract this from that

00:04:27.952 --> 00:04:29.526
and so what do I get?

00:04:29.526 --> 00:04:30.516
Well, let's see.

00:04:30.516 --> 00:04:33.063
I'm gonna have one minus p

00:04:33.063 --> 00:04:35.908
and then if I subtract
1p times one minus p

00:04:35.908 --> 00:04:38.023
from 2p times one minus p,

00:04:38.023 --> 00:04:41.606
well I'm just going to
be left with plus 1p

00:04:42.548 --> 00:04:43.965
times one minus p

00:04:45.689 --> 00:04:47.830
and then if I subtract this from that,

00:04:47.830 --> 00:04:51.997
I'm gonna be left with 1p
times one minus p squared

00:04:54.416 --> 00:04:57.152
and we're just gonna keep
going on and on and on

00:04:57.152 --> 00:04:59.380
and so let me simplify this a little bit.

00:04:59.380 --> 00:05:01.012
If I distribute this negative,

00:05:01.012 --> 00:05:05.142
this could be plus and then
this would be p minus one

00:05:05.142 --> 00:05:07.823
and then if we distribute
this expected value of X,

00:05:07.823 --> 00:05:10.249
we get on the left-hand side the,

00:05:10.249 --> 00:05:11.868
and let me scroll up a little bit.

00:05:11.868 --> 00:05:14.195
I don't want to scrunch it too much.

00:05:14.195 --> 00:05:17.536
So let's see, we have
the expected value of X

00:05:17.536 --> 00:05:21.693
and then plus p times
the expected value of X.

00:05:21.693 --> 00:05:23.799
P times the expected value of X

00:05:23.799 --> 00:05:26.577
minus the expected value of X,

00:05:26.577 --> 00:05:28.106
these cancel out,

00:05:28.106 --> 00:05:32.273
is going to be equal to p
plus p times one minus p

00:05:34.416 --> 00:05:37.083
plus p times one minus p squared

00:05:38.461 --> 00:05:40.768
and it's gonna keep
going on and on and on.

00:05:40.768 --> 00:05:42.573
Well, on the left-hand side all I have

00:05:42.573 --> 00:05:44.176
is a p times expected value of X.

00:05:44.176 --> 00:05:45.923
If I want to solve for
the expected value of X,

00:05:45.923 --> 00:05:48.056
I just divide both sides by p.

00:05:48.056 --> 00:05:49.998
So I get and this is kind of neat

00:05:49.998 --> 00:05:51.986
through this mathematical gymnastics,

00:05:51.986 --> 00:05:55.341
I now have, I'm just dividing
everything by p, both sides,

00:05:55.341 --> 00:05:57.890
on the left-hand side I just
have the expected value of X.

00:05:57.890 --> 00:05:59.913
If I divide all of these terms by p,

00:05:59.913 --> 00:06:01.225
this first term becomes one,

00:06:01.225 --> 00:06:04.225
the second term becomes one minus p,

00:06:05.521 --> 00:06:07.392
this third term, if I divide by p,

00:06:07.392 --> 00:06:11.559
becomes plus one minus p
squared, so forth and so on.

00:06:13.007 --> 00:06:14.466
Now what's cool about this,

00:06:14.466 --> 00:06:17.350
this is a classic geometric series

00:06:17.350 --> 00:06:20.017
with a common ratio of one minus p

00:06:20.017 --> 00:06:22.407
and if that term is
completely unfamiliar to you,

00:06:22.407 --> 00:06:24.216
I encourage you and this
is why it's actually called

00:06:24.216 --> 00:06:25.686
a geometric, one of the reasons,

00:06:25.686 --> 00:06:28.232
arguments for why it's called
a geometric random variable,

00:06:28.232 --> 00:06:31.031
but I encourage you to review
what a geometric series is

00:06:31.031 --> 00:06:33.631
on Khan Academy if this
looks completely unfamiliar,

00:06:33.631 --> 00:06:35.520
but in other places we proved

00:06:35.520 --> 00:06:38.067
using actually a very similar
technique that we did up here

00:06:38.067 --> 00:06:40.467
that this sum is going to be equal to

00:06:40.467 --> 00:06:43.969
one over one minus our common ratio

00:06:43.969 --> 00:06:47.748
and our common ratio is one minus p.

00:06:47.748 --> 00:06:49.638
So what is this going to be equal to?

00:06:49.638 --> 00:06:52.643
And we are really in the
home stretch right over here.

00:06:52.643 --> 00:06:56.810
This is going to be equal to
one over one minus one plus p.

00:06:58.048 --> 00:07:00.440
One minus one plus p.

00:07:00.440 --> 00:07:03.692
Which is indeed equal to one over p.

00:07:03.692 --> 00:07:06.309
So there you have it, we
have proven to ourselves

00:07:06.309 --> 00:07:10.116
that the expected value of
a geometric random variable

00:07:10.116 --> 00:07:13.562
using some, I think, cool mathematics

00:07:13.562 --> 00:07:16.062
is indeed equal to one over p.


Cumulative geometric probability (less than a value)

WEBVTT
Kind: captions
Language: en

00:00:00.653 --> 00:00:03.284
- [Instructor] Lilyana runs
a cake decorating business,

00:00:03.284 --> 00:00:07.200
for which 10% of her orders
come over the telephone.

00:00:07.200 --> 00:00:10.835
Let C be the number of cake
orders Lilyana receives

00:00:10.835 --> 00:00:15.002
in a month until she first gets
an order over the telephone.

00:00:16.544 --> 00:00:21.529
Assume the method of placing
each cake order is independent.

00:00:21.529 --> 00:00:24.033
So C, if we assume a few things,

00:00:24.033 --> 00:00:27.283
is a classic geometric random variable.

00:00:28.784 --> 00:00:29.617
What tells us that?

00:00:29.617 --> 00:00:31.254
Well, a giveaway is that we're gonna keep

00:00:31.254 --> 00:00:33.813
doing these independent trials,

00:00:33.813 --> 00:00:36.302
where the probability
of success is constant,

00:00:36.302 --> 00:00:37.794
and there's a clear success.

00:00:37.794 --> 00:00:40.030
A telephone order in
this case is a success.

00:00:40.030 --> 00:00:42.171
The probability is 10% of it happening.

00:00:42.171 --> 00:00:46.267
And we're gonna keep doing
it until we get a success.

00:00:46.267 --> 00:00:48.832
So classic geometric random variable.

00:00:48.832 --> 00:00:52.397
Now they ask us, find the
probability, the probability,

00:00:52.397 --> 00:00:54.176
that it takes fewer than five orders

00:00:54.176 --> 00:00:56.702
for Lilyana to get her first
telephone order of the month.

00:00:56.702 --> 00:01:00.943
So it's really the probability
that C is less than five.

00:01:00.943 --> 00:01:04.033
So like always, pause this
video and have a go at it.

00:01:04.033 --> 00:01:06.655
And even if you struggle
with it, that's better.

00:01:06.655 --> 00:01:09.991
Your brain will be more
primed for the actual solution

00:01:09.991 --> 00:01:11.918
that we can go through together.

00:01:11.918 --> 00:01:13.105
Alright. (chuckles)

00:01:13.105 --> 00:01:14.760
So I'm assuming you've had a go at it.

00:01:14.760 --> 00:01:16.135
So there's a couple of
ways to approach it.

00:01:16.135 --> 00:01:17.672
You could say, well, look,
this is just gonna be

00:01:17.672 --> 00:01:20.742
the probability that C is equal to one,

00:01:20.742 --> 00:01:23.705
plus the probability
that C is equal to two,

00:01:23.705 --> 00:01:26.758
plus the probability
that C is equal to three,

00:01:26.758 --> 00:01:30.302
plus the probability
that C is equal to four.

00:01:30.302 --> 00:01:32.227
and we can calculate it this way.

00:01:32.227 --> 00:01:34.787
What is the probability that C equals one?

00:01:34.787 --> 00:01:36.816
Well, it's the probability
that her very first order

00:01:36.816 --> 00:01:38.284
is a telephone order.

00:01:38.284 --> 00:01:40.117
And so we'll have 0.1.

00:01:40.979 --> 00:01:42.861
What's the probability that C equals two?

00:01:42.861 --> 00:01:44.024
Well, it's the probability
that her first order

00:01:44.024 --> 00:01:45.685
is not a telephone order.

00:01:45.685 --> 00:01:47.231
So it's one minus 10%.

00:01:47.231 --> 00:01:49.867
There's a 90% chance it's
not a telephone order,

00:01:49.867 --> 00:01:52.960
and that her second order
is a telephone order.

00:01:52.960 --> 00:01:54.906
What about the probability C equals three?

00:01:54.906 --> 00:01:59.711
Well, her first two orders
would not be telephone orders,

00:01:59.711 --> 00:02:02.118
and her third order would be one.

00:02:02.118 --> 00:02:03.426
And then C equals four?

00:02:03.426 --> 00:02:07.887
Well, her first three orders
would not be telephone orders,

00:02:07.887 --> 00:02:09.999
and her fourth one would.

00:02:09.999 --> 00:02:11.778
And we could get a calculator maybe,

00:02:11.778 --> 00:02:14.010
and add all of these things up,

00:02:14.010 --> 00:02:16.244
and we would actually get the answer.

00:02:16.244 --> 00:02:17.220
But you're probably wondering,

00:02:17.220 --> 00:02:19.602
well, this is kind of hairy
to type into a calculator.

00:02:19.602 --> 00:02:22.477
Maybe there is an easier
way to tackle this.

00:02:22.477 --> 00:02:24.733
And indeed, there is.

00:02:24.733 --> 00:02:25.851
So think about it.

00:02:25.851 --> 00:02:28.592
The probability that C is less than five,

00:02:28.592 --> 00:02:32.759
that's the same thing as
one minus the probability

00:02:34.008 --> 00:02:37.651
that we don't have a telephone
order in the first four.

00:02:37.651 --> 00:02:41.234
One minus the probability
that no telephone

00:02:43.582 --> 00:02:44.415
order

00:02:45.447 --> 00:02:46.280
in first

00:02:48.267 --> 00:02:49.267
four orders.

00:02:52.401 --> 00:02:54.054
So what's this?

00:02:54.054 --> 00:02:55.777
Well, 'cause this is just saying,

00:02:55.777 --> 00:02:56.610
what's the probability we do

00:02:56.610 --> 00:02:58.292
have an order in the first four?

00:02:58.292 --> 00:03:00.133
So it's the same thing as
one minus the probability

00:03:00.133 --> 00:03:02.266
that we don't have an
order in the first four.

00:03:02.266 --> 00:03:04.987
And this is pretty
straightforward to calculate.

00:03:04.987 --> 00:03:07.451
So this is going to be equal to one minus,

00:03:07.451 --> 00:03:08.603
and let me do this in another color

00:03:08.603 --> 00:03:10.222
so we know what I'm referring to.

00:03:10.222 --> 00:03:13.112
So what's the probability that
we have no telephone orders

00:03:13.112 --> 00:03:14.950
in the first four orders?

00:03:14.950 --> 00:03:16.429
Well, the probability on a given order

00:03:16.429 --> 00:03:19.499
that you don't have a
telephone order is 0.9.

00:03:19.499 --> 00:03:21.677
And then if that has to be
true for the first four,

00:03:21.677 --> 00:03:25.296
well, it's gonna be 0.9
times 0.9 times 09 times 0.9,

00:03:25.296 --> 00:03:27.644
or 0.9 to the fourth power.

00:03:27.644 --> 00:03:30.385
So this a lot easier to
calculate, so let's do that.

00:03:30.385 --> 00:03:32.277
Let's get a calculator out.

00:03:32.277 --> 00:03:36.444
Alright, so let me just
take .9 to the fourth power

00:03:37.405 --> 00:03:40.210
is equal to, and then let
me subtract that from one.

00:03:40.210 --> 00:03:42.016
So let me make that a negative,

00:03:42.016 --> 00:03:43.735
and then let me add one to it.

00:03:43.735 --> 00:03:46.485
And we get, there you go, 0.3439.

00:03:48.535 --> 00:03:50.118
So this is equal to

00:03:51.923 --> 00:03:52.756
0.3439.

00:03:57.415 --> 00:03:58.570
And we're done.

00:03:58.570 --> 00:04:01.804
That's the probability that it
takes fewer than five orders

00:04:01.804 --> 00:04:05.826
for her to get her first
telephone order of the month.


Cumulative geometric probability (greater than a value)

WEBVTT
Kind: captions
Language: en

00:00:00.837 --> 00:00:02.272
- [Instructor] Emelia
registers vehicles for

00:00:02.272 --> 00:00:04.530
the Department of Transportation.

00:00:04.530 --> 00:00:07.527
Sports utility vehicles,
also known as SUVs,

00:00:07.527 --> 00:00:11.736
make up 12% of the vehicles she registers.

00:00:11.736 --> 00:00:15.046
Let V be the number of vehicles Emelia

00:00:15.046 --> 00:00:19.213
registers in a day until
she first registers an SUV.

00:00:21.049 --> 00:00:25.093
Assume the type of each
vehicle is independent.

00:00:25.093 --> 00:00:28.118
Find the probability that Emelia registers

00:00:28.118 --> 00:00:30.810
more than four, more than four,

00:00:30.810 --> 00:00:33.893
vehicles before she registers an SUV.

00:00:35.834 --> 00:00:36.837
So, let's just first think about

00:00:36.837 --> 00:00:39.349
what this random variable V is.

00:00:39.349 --> 00:00:43.173
So, it's the number of vehicles
Emelia registers in a day,

00:00:43.173 --> 00:00:45.423
until she registers an SUV.

00:00:46.640 --> 00:00:48.836
So, for example if the
first person who walks

00:00:48.836 --> 00:00:50.668
in the line or through the door,

00:00:50.668 --> 00:00:52.790
has an SUV and they're
trying to register it,

00:00:52.790 --> 00:00:54.666
then V would be equal to one.

00:00:54.666 --> 00:00:56.513
If the first person isn't an SUV,

00:00:56.513 --> 00:00:59.399
but the second person is,
then V would be equal to two,

00:00:59.399 --> 00:01:00.967
so forth and so on.

00:01:00.967 --> 00:01:03.707
So, this right over here
is a classic geometric

00:01:03.707 --> 00:01:06.457
random variable, right over here.

00:01:08.369 --> 00:01:10.431
So, geometric random variable.

00:01:10.431 --> 00:01:14.539
We have a very clear success
metric for each trial.

00:01:14.539 --> 00:01:16.817
Do we have a SUV or not?

00:01:16.817 --> 00:01:19.341
Each trial is independent,
they tell us that.

00:01:19.341 --> 00:01:20.783
They are independent.

00:01:20.783 --> 00:01:23.308
The probability of success
in each trial is constant.

00:01:23.308 --> 00:01:27.268
We have a 12% success for each new person

00:01:27.268 --> 00:01:28.962
who comes through the line.

00:01:28.962 --> 00:01:31.857
Now the reason this is not
a binomial random variable,

00:01:31.857 --> 00:01:35.372
is that we do not have a
finite number of trials.

00:01:35.372 --> 00:01:38.058
Here, we're gonna keep performing trials.

00:01:38.058 --> 00:01:40.814
We're gonna keep serving
people in the line,

00:01:40.814 --> 00:01:42.481
until we get an SUV.

00:01:44.550 --> 00:01:46.690
And so, what we have over here,

00:01:46.690 --> 00:01:48.668
when they say find the probability

00:01:48.668 --> 00:01:50.543
that Emelia registers
more than four vehicles

00:01:50.543 --> 00:01:52.186
before she registers an SUV.

00:01:52.186 --> 00:01:56.353
This is the probability
that V is greater than four.

00:01:58.011 --> 00:01:59.922
So, I encourage you like always,

00:01:59.922 --> 00:02:02.467
pause this video and see
if you can work through it.

00:02:02.467 --> 00:02:04.819
And we're gonna assume, she's
not just gonna leave her,

00:02:04.819 --> 00:02:07.491
I guess her desk, or wherever the things

00:02:07.491 --> 00:02:09.096
are being registered.

00:02:09.096 --> 00:02:10.267
She's not going to leave the counter

00:02:10.267 --> 00:02:14.034
until someone shows up registering an SUV.

00:02:14.034 --> 00:02:17.944
So, we will just keep looking at people,

00:02:17.944 --> 00:02:20.182
I guess we could say over
multiple days, forever.

00:02:20.182 --> 00:02:22.532
She'll work for an
infinite number of years,

00:02:22.532 --> 00:02:24.188
just for the sake of this problem,

00:02:24.188 --> 00:02:27.364
until an SUV actually shows up.

00:02:27.364 --> 00:02:29.127
So, try to figure this out.

00:02:29.127 --> 00:02:31.274
Alright, I'm assuming you've had a go

00:02:31.274 --> 00:02:33.584
and some of you might said,
well, isn't this going to be

00:02:33.584 --> 00:02:36.345
equal to the probability
that V is equal to five,

00:02:36.345 --> 00:02:39.910
plus the probability
that V is equal to six,

00:02:39.910 --> 00:02:42.953
plus the probability
that V is equal to seven,

00:02:42.953 --> 00:02:45.544
and it just goes on and on and on forever.

00:02:45.544 --> 00:02:47.557
And this is actually true.

00:02:47.557 --> 00:02:49.325
And you say, well, how
do I calculate this?

00:02:49.325 --> 00:02:51.786
I'm just summing up an
infinite number of things.

00:02:51.786 --> 00:02:55.953
Now the key realization here,
is that one way to think

00:02:57.042 --> 00:03:00.593
about the probability that
V is greater than four,

00:03:00.593 --> 00:03:03.100
is this is the same
thing as the probability

00:03:03.100 --> 00:03:06.517
that V is not less than or equal to four,

00:03:12.088 --> 00:03:14.681
these two things are equivalent.

00:03:14.681 --> 00:03:16.645
So, what's the probability that V

00:03:16.645 --> 00:03:19.701
is not less than or equal to four?

00:03:19.701 --> 00:03:21.851
This might be a slightly easier
thing for you to calculate.

00:03:21.851 --> 00:03:22.943
Once again, pause the
video and see if you can

00:03:22.943 --> 00:03:24.246
figure it out.

00:03:24.246 --> 00:03:27.134
Well what's the probability
that V is not less than

00:03:27.134 --> 00:03:28.411
or equal to four?

00:03:28.411 --> 00:03:31.575
Well that's the same thing
as the probability of

00:03:31.575 --> 00:03:35.742
first four customers, or
first four, I guess people.

00:03:38.186 --> 00:03:41.718
First four, I'll say customers,
or I'll say first four cars.

00:03:41.718 --> 00:03:44.218
The customer's cars, not SUVs.

00:03:46.647 --> 00:03:48.792
So, this one is feeling
pretty straightforward.

00:03:48.792 --> 00:03:52.041
What's the probability that
for each customer she goes to,

00:03:52.041 --> 00:03:53.480
that they're not an SUV?

00:03:53.480 --> 00:03:57.647
Well that's one minus
12%, or 88%, or 0.88.

00:03:58.577 --> 00:04:00.158
And if we want to know the probability

00:04:00.158 --> 00:04:02.158
of the first four cars are not SUVs.

00:04:02.158 --> 00:04:05.648
Well that's 0.88 to the fourth power.

00:04:05.648 --> 00:04:07.195
And so that's all we have to calculate.

00:04:07.195 --> 00:04:09.866
And so let's get our calculator out.

00:04:09.866 --> 00:04:13.030
And say I'm going to get, oops.

00:04:13.030 --> 00:04:17.186
I'm going to get 0.88 and
I'm going to raise it to

00:04:17.186 --> 00:04:19.278
the fourth power and I get,

00:04:19.278 --> 00:04:22.346
and I'm just going to round it to,

00:04:22.346 --> 00:04:25.127
the nearest, let's see, do
they tell me to round it?

00:04:25.127 --> 00:04:29.643
Okay, I'll just round it to
the nearest, I guess 100th,

00:04:29.643 --> 00:04:32.560
well, I'll just write it as 0.5997.

00:04:35.270 --> 00:04:38.103
Is approximately equal to, 0.5997.

00:04:42.497 --> 00:04:44.585
If you wanted to write
this as a percentage

00:04:44.585 --> 00:04:47.752
it would be approximately 59.97%.

00:04:50.033 --> 00:04:52.109
So a little bit better than half,

00:04:52.109 --> 00:04:55.545
than a 50% shot, a little
less than a two-thirds shot,

00:04:55.545 --> 00:04:59.540
that she is going to have to
see more than four customers

00:04:59.540 --> 00:05:01.373
until she sees an SUV.


Probability for a geometric random variable

WEBVTT
Kind: captions
Language: en

00:00:00.608 --> 00:00:03.497
- [Instructor] Jeremiah makes 25%

00:00:03.497 --> 00:00:05.668
of the three-point shots he attempts.

00:00:05.668 --> 00:00:07.562
Far better than my percentage.

00:00:07.562 --> 00:00:10.046
For warm up, Jeremiah likes to shoot

00:00:10.046 --> 00:00:13.829
three-point shots until
he successfully makes one.

00:00:13.829 --> 00:00:15.383
Alright, this is the telltale signs

00:00:15.383 --> 00:00:17.345
of a geometric random variables.

00:00:17.345 --> 00:00:18.947
How many trials do I have to take

00:00:18.947 --> 00:00:21.430
until I get a success?

00:00:21.430 --> 00:00:25.263
Let M be the number of
shots it takes Jeremiah

00:00:26.126 --> 00:00:29.070
to successfully make his
first three-point shot.

00:00:29.070 --> 00:00:31.273
Okay, so they're defining
the random variable here,

00:00:31.273 --> 00:00:33.037
the number of shots it takes,

00:00:33.037 --> 00:00:34.638
the number of trials it takes

00:00:34.638 --> 00:00:38.716
until we get a successful
three-point shot.

00:00:38.716 --> 00:00:42.424
Assume that the results of
each shot are independent.

00:00:42.424 --> 00:00:44.472
Alright, the probability
that he makes a given shot

00:00:44.472 --> 00:00:46.424
is not dependent on
whether he made or missed

00:00:46.424 --> 00:00:48.036
the previous shots.

00:00:48.036 --> 00:00:52.724
Find the probability that
Jeremiah's first successful shot

00:00:52.724 --> 00:00:55.344
occurs on his third attempt.

00:00:55.344 --> 00:00:57.307
So, like always, pause this video

00:00:57.307 --> 00:00:59.979
and see if you can have a go at it.

00:00:59.979 --> 00:01:01.719
Alright, now let's work
through this together.

00:01:01.719 --> 00:01:04.759
So, we wanna find the probability.

00:01:04.759 --> 00:01:06.584
So, M is the number of shots it takes

00:01:06.584 --> 00:01:09.568
until Jeremiah makes
his first successful one

00:01:09.568 --> 00:01:11.169
and so, what they're really asking us,

00:01:11.169 --> 00:01:14.145
find the probability
that M is equal to three,

00:01:14.145 --> 00:01:15.899
that his first successful shot

00:01:15.899 --> 00:01:18.200
occurs on his third attempt.

00:01:18.200 --> 00:01:20.200
So, M is equal to three.

00:01:22.406 --> 00:01:25.425
So, that the number of
shots it takes Jeremiah

00:01:25.425 --> 00:01:29.244
to make his first
successful shot is three.

00:01:29.244 --> 00:01:31.396
So, how do we do this?

00:01:31.396 --> 00:01:34.949
Well, what's the probability
of that happening?

00:01:34.949 --> 00:01:39.020
Well, that means he has to
miss his first two shots

00:01:39.020 --> 00:01:42.003
and then make his third shot.

00:01:42.003 --> 00:01:46.262
So, what's the probability of
him missing his first shot?

00:01:46.262 --> 00:01:48.480
Well, if he has a 1/4 chance

00:01:48.480 --> 00:01:53.294
of making his shots, he
has a 3/4 probability

00:01:53.294 --> 00:01:55.005
of missing his shots,

00:01:55.005 --> 00:01:56.931
so this will be 3/4,

00:01:56.931 --> 00:01:59.034
so he misses the first shot,

00:01:59.034 --> 00:02:01.891
times, he has to miss the second shot

00:02:01.891 --> 00:02:05.141
and then he has to make his third shot,

00:02:07.620 --> 00:02:08.994
so there you have it.

00:02:08.994 --> 00:02:10.655
That's the probability.

00:02:10.655 --> 00:02:11.988
Miss, miss, make

00:02:13.077 --> 00:02:15.166
and so, what is this going to be?

00:02:15.166 --> 00:02:17.916
This is equal to nine over 64ths.

00:02:19.639 --> 00:02:20.472
So, there you have.

00:02:20.472 --> 00:02:23.266
If you wanted to have this as a decimal,

00:02:23.266 --> 00:02:25.358
we could get a calculator out real fast,

00:02:25.358 --> 00:02:27.775
so this is nine divided by 64

00:02:30.604 --> 00:02:32.687
is equal to roughly 0.14.

00:02:33.797 --> 00:02:35.297
Approximately 0.14

00:02:36.291 --> 00:02:37.487
or another way to think about it

00:02:37.487 --> 00:02:42.406
is a roughly 14% chance or 14% probability

00:02:42.406 --> 00:02:44.906
that his first successful shot

00:02:46.323 --> 00:02:48.656
occurs in his third attempt.


Geometric random variables introduction

WEBVTT
Kind: captions
Language: en

00:00:01.629 --> 00:00:04.554
- [Narrator] So I have two,
different random variables here.

00:00:04.554 --> 00:00:06.370
And what I wanna do is think about

00:00:06.370 --> 00:00:08.404
what type of random variables they are.

00:00:08.404 --> 00:00:10.803
So this first random variable, x,

00:00:10.803 --> 00:00:12.818
is equal to the number of sixes

00:00:12.818 --> 00:00:15.235
after 12 rolls of a fair die.

00:00:16.130 --> 00:00:18.308
Well this looks pretty much like

00:00:18.308 --> 00:00:20.424
a binomial random variable.

00:00:20.424 --> 00:00:21.969
In fact, I'm pretty confident it is

00:00:21.969 --> 00:00:23.515
a binomial random variable and we

00:00:23.515 --> 00:00:25.822
can just go down the checklist.

00:00:25.822 --> 00:00:27.612
The outcome of each trial can be

00:00:27.612 --> 00:00:29.572
a success or failure.

00:00:29.572 --> 00:00:32.655
So, trial outcome success or failure.

00:00:39.760 --> 00:00:41.831
It's either gonna go either way.

00:00:41.831 --> 00:00:44.068
The result of each trial is independent

00:00:44.068 --> 00:00:45.192
from the other one.

00:00:45.192 --> 00:00:47.557
Whether I get a six on the third trial

00:00:47.557 --> 00:00:48.775
is independent of whether I got

00:00:48.775 --> 00:00:51.147
a six on the first or the second trial.

00:00:51.147 --> 00:00:53.351
So result, let me write this,

00:00:53.351 --> 00:00:55.841
trial, I'll just do a shorthand trial,

00:00:55.841 --> 00:00:58.591
results independent, independent,

00:01:01.598 --> 00:01:03.539
that's an important condition.

00:01:03.539 --> 00:01:07.797
Let's see, there are a
fixed number of trials.

00:01:07.797 --> 00:01:09.714
Fixed number of trials.

00:01:12.825 --> 00:01:15.925
In this case we're gonna have 12 trials.

00:01:15.925 --> 00:01:17.701
And then the last one is, we have

00:01:17.701 --> 00:01:20.457
the same probability on each trial.

00:01:20.457 --> 00:01:22.707
Same probability of success

00:01:23.923 --> 00:01:26.090
probability on each trial.

00:01:30.358 --> 00:01:33.553
So yes indeed, this met
all of the conditions

00:01:33.553 --> 00:01:37.470
for being a binomial,
binomial random variable.

00:01:43.336 --> 00:01:44.355
And this was all just a little

00:01:44.355 --> 00:01:45.687
bit of review about things that

00:01:45.687 --> 00:01:48.030
we have talked about in other videos.

00:01:48.030 --> 00:01:49.771
But what about this thing
in the salmon color?

00:01:49.771 --> 00:01:51.944
The random variable y.

00:01:51.944 --> 00:01:54.480
So this says the number of rolls

00:01:54.480 --> 00:01:57.230
until we get a six on a fair die.

00:01:58.500 --> 00:02:00.703
So this one strikes us as
a little bit different.

00:02:00.703 --> 00:02:03.433
But let's see where it
is actually different.

00:02:03.433 --> 00:02:06.855
So, does it meet that the trial outcomes

00:02:06.855 --> 00:02:10.665
that there's a clear success
or failure for each trial?

00:02:10.665 --> 00:02:12.328
Well yeah, we're just gonna keep rolling.

00:02:12.328 --> 00:02:14.474
So each time we roll, it's a trial.

00:02:14.474 --> 00:02:16.092
And success is when we get a six.

00:02:16.092 --> 00:02:18.406
Failure is when we don't get a six.

00:02:18.406 --> 00:02:20.113
So the outcome of each trial can

00:02:20.113 --> 00:02:23.520
be classified as either
a success or a failure.

00:02:23.520 --> 00:02:25.846
So it meets, maybe I'll put the checks

00:02:25.846 --> 00:02:29.011
right over here, it meets
this first constraint.

00:02:29.011 --> 00:02:32.612
Are the results of each trial independent?

00:02:32.612 --> 00:02:34.857
Well whether I get a six on the first roll

00:02:34.857 --> 00:02:36.971
or the second roll, or the third roll,

00:02:36.971 --> 00:02:39.134
or the fourth roll, or the third roll,

00:02:39.134 --> 00:02:41.341
the probabilities shouldn't be dependent

00:02:41.341 --> 00:02:45.181
on whether I did or didn't
get a six on a previous roll.

00:02:45.181 --> 00:02:47.598
So, we have the independents.

00:02:48.494 --> 00:02:49.866
And we also have the same probability

00:02:49.866 --> 00:02:51.137
of success on each trial.

00:02:51.137 --> 00:02:53.161
In every case it's a 1/6 probability that

00:02:53.161 --> 00:02:55.935
I get a six, so this stays constant.

00:02:55.935 --> 00:02:58.557
And I skipped this third
condition for a reason.

00:02:58.557 --> 00:03:02.832
Because we clearly don't have
a fixed number of trials.

00:03:02.832 --> 00:03:07.558
Over here we could roll 50
times until we get a six.

00:03:07.558 --> 00:03:08.609
The probability that we'd have

00:03:08.609 --> 00:03:10.424
to roll 50 times is very low.

00:03:10.424 --> 00:03:12.227
But we might have to roll 500 times

00:03:12.227 --> 00:03:13.619
in order to get a six.

00:03:13.619 --> 00:03:16.694
In fact, think about what
the minimum value of y is

00:03:16.694 --> 00:03:19.576
and what the maximum value of y is.

00:03:19.576 --> 00:03:23.743
So the minimum value that
this random variable can take,

00:03:24.992 --> 00:03:27.704
I'll just call it min y, is equal to what?

00:03:27.704 --> 00:03:29.263
Well, it's gonna take at least one roll.

00:03:29.263 --> 00:03:31.064
So that's the minimum value.

00:03:31.064 --> 00:03:34.309
But what is the maximum value for y?

00:03:34.309 --> 00:03:36.468
And I'll let you think about that.

00:03:36.468 --> 00:03:39.198
I'll assumed you thought about
it, if you paused the video.

00:03:39.198 --> 00:03:41.024
Well, there is no max value.

00:03:41.024 --> 00:03:42.677
You can't say, "Oh it's a billion."

00:03:42.677 --> 00:03:44.401
Because there's some probability

00:03:44.401 --> 00:03:46.862
that it might take a
billion and one rolls.

00:03:46.862 --> 00:03:49.413
It is a very, very, very,
very, very, very small

00:03:49.413 --> 00:03:51.977
probability, but there's some probability.

00:03:51.977 --> 00:03:55.786
It could take a Google
rolls, a Google plex rolls.

00:03:55.786 --> 00:03:58.321
So you can imagine where this is going.

00:03:58.321 --> 00:04:01.020
So this type of random variable,

00:04:01.020 --> 00:04:03.393
where it meets a lot of the constraints

00:04:03.393 --> 00:04:05.043
of a binomial random variable.

00:04:05.043 --> 00:04:08.025
Each trial has a clear
success or failure outcome.

00:04:08.025 --> 00:04:10.740
The probability of success
on each trial is constant.

00:04:10.740 --> 00:04:14.158
The trial results are
independent of each other.

00:04:14.158 --> 00:04:16.215
But we don't have a
fixed number of trials.

00:04:16.215 --> 00:04:17.746
In fact, it's a situation, we're saying,

00:04:17.746 --> 00:04:19.906
"How many trials do we need to get,

00:04:19.906 --> 00:04:23.459
"to we need to have until we get success?"

00:04:23.459 --> 00:04:25.458
Maybe that's a general way of framing

00:04:25.458 --> 00:04:27.378
this type of random variable.

00:04:27.378 --> 00:04:29.878
How many trials until success?

00:04:38.371 --> 00:04:40.892
While the binomial random variable was,

00:04:40.892 --> 00:04:44.142
how many trials, or how many successes,

00:04:47.988 --> 00:04:50.905
I should say, how many successes in

00:04:52.748 --> 00:04:54.748
finite number of trials?

00:04:59.257 --> 00:05:00.863
So if you see this general form

00:05:00.863 --> 00:05:02.226
and it meets these conditions, you can

00:05:02.226 --> 00:05:05.026
feel good it's a binomial random variable.

00:05:05.026 --> 00:05:07.207
But if we're meeting this condition,

00:05:07.207 --> 00:05:09.532
clear success or failure outcome,

00:05:09.532 --> 00:05:12.170
independent trials, constant probability,

00:05:12.170 --> 00:05:13.790
but we're not talking about the successes

00:05:13.790 --> 00:05:15.470
in a finite number of trials.

00:05:15.470 --> 00:05:18.426
We're talking about how
many trials until success?

00:05:18.426 --> 00:05:20.438
Then this type of random variable

00:05:20.438 --> 00:05:23.605
is called a geometric random variable.

00:05:28.736 --> 00:05:30.922
And we will see why, in future videos

00:05:30.922 --> 00:05:32.839
it is called geometric.

00:05:34.056 --> 00:05:35.902
Because the math that involves

00:05:35.902 --> 00:05:38.331
the probabilities of various outcomes

00:05:38.331 --> 00:05:41.044
looks a lot like geometric growth,

00:05:41.044 --> 00:05:43.103
or geometric sequences and series

00:05:43.103 --> 00:05:45.563
that we look at in other
types of mathematics.

00:05:45.563 --> 00:05:46.930
And in case I forgot to mention,

00:05:46.930 --> 00:05:48.280
the reason why they're called binomial

00:05:48.280 --> 00:05:50.153
random variables is because when you

00:05:50.153 --> 00:05:52.825
think about the probabilities
of different outcomes,

00:05:52.825 --> 00:05:55.137
you have these things called
binomial coefficients,

00:05:55.137 --> 00:05:56.993
based on combinatorics.

00:05:56.993 --> 00:05:58.899
And those come out of things like

00:05:58.899 --> 00:06:01.048
Pascal's Triangle and when you take

00:06:01.048 --> 00:06:03.881
a binomial to ever increasing powers.

00:06:03.881 --> 00:06:06.135
So that's where those words come from.

00:06:06.135 --> 00:06:07.602
But in the next few videos, the important

00:06:07.602 --> 00:06:10.092
thing is to recognize the
difference between the two.

00:06:10.092 --> 00:06:11.306
And then we're gonna start thinking

00:06:11.306 --> 00:06:14.743
about how do we deal with
geometric random variables.


Finding the mean and standard deviation of a binomial random variable

WEBVTT
Kind: captions
Language: en

00:00:00.746 --> 00:00:02.805
- [Instructor] We're
told a company produces

00:00:02.805 --> 00:00:05.042
processing chips for cell phones.

00:00:05.042 --> 00:00:07.085
At one of its large factories,

00:00:07.085 --> 00:00:11.195
two percent of the chips produced
are defective in some way.

00:00:11.195 --> 00:00:13.668
A quality check involves
randomly selecting

00:00:13.668 --> 00:00:15.501
and testing 500 chips.

00:00:16.634 --> 00:00:20.709
What are the mean and standard
deviation of the number

00:00:20.709 --> 00:00:24.079
of defective processing
chips in these samples?

00:00:24.079 --> 00:00:26.264
Like always, try to pause this video

00:00:26.264 --> 00:00:28.145
and have a go at it on your own,

00:00:28.145 --> 00:00:30.551
and then we will work through it together.

00:00:30.551 --> 00:00:33.895
All right, so let me
define a random variable

00:00:33.895 --> 00:00:36.524
that we're gonna find the mean
and standard deviation of,

00:00:36.524 --> 00:00:37.661
and I'm gonna make that random variable

00:00:37.661 --> 00:00:39.623
the number of defective processing chips

00:00:39.623 --> 00:00:41.373
in a 500-chip sample.

00:00:42.894 --> 00:00:47.061
Let's let x be equal to the
number of defective chips ...

00:00:53.224 --> 00:00:54.807
in 500-chip sample.

00:00:59.939 --> 00:01:02.287
The first thing to recognize is that

00:01:02.287 --> 00:01:05.607
this will be a binomial variable.

00:01:05.607 --> 00:01:07.024
This is binomial.

00:01:08.341 --> 00:01:10.178
How do we know it's binomial?

00:01:10.178 --> 00:01:12.345
Well, it's made up of 500,

00:01:13.582 --> 00:01:17.058
it's a finite number of
trials right over here.

00:01:17.058 --> 00:01:19.281
The probability of
getting a defective chip,

00:01:19.281 --> 00:01:21.128
you could do this as a
probability of success.

00:01:21.128 --> 00:01:22.584
It's a little bit counterintuitive

00:01:22.584 --> 00:01:24.616
that a defective chip would be a success,

00:01:24.616 --> 00:01:26.830
but we're summing up the defective chips,

00:01:26.830 --> 00:01:31.126
so we would view the
probability of a defect,

00:01:31.126 --> 00:01:33.959
or I should say, a defective chip,

00:01:34.957 --> 00:01:38.207
it is constant across these 500 trials,

00:01:39.141 --> 00:01:41.682
and we will assume that they
are independent of each other,

00:01:41.682 --> 00:01:42.515
0.02.

00:01:43.662 --> 00:01:45.807
You might be saying, "Hey,
well, are we replacing the chips

00:01:45.807 --> 00:01:47.411
"before or after?" but we're assuming

00:01:47.411 --> 00:01:50.459
it's from a functionally
infinite population,

00:01:50.459 --> 00:01:52.174
or if you want to make it feel better,

00:01:52.174 --> 00:01:53.872
you could say, maybe you
are replacing the chips.

00:01:53.872 --> 00:01:55.954
They're not really telling
us that right over here,

00:01:55.954 --> 00:01:57.803
so we'll assume that each of these trials

00:01:57.803 --> 00:01:59.388
are independent of each other,

00:01:59.388 --> 00:02:00.728
and that the probability of getting

00:02:00.728 --> 00:02:04.078
a defective chip stays constant here.

00:02:04.078 --> 00:02:06.356
So this is a binomial random variable,

00:02:06.356 --> 00:02:08.998
or binomial variable,
and we know the formulas

00:02:08.998 --> 00:02:10.899
for the mean and standard deviation

00:02:10.899 --> 00:02:12.695
of a binomial variable.

00:02:12.695 --> 00:02:16.970
The mean, the mean of x,
which is the same thing

00:02:16.970 --> 00:02:19.687
as the expected value of x,

00:02:19.687 --> 00:02:24.288
is going to be equal to
the number of trials, n,

00:02:24.288 --> 00:02:28.668
times the probability of
a success on each trial,

00:02:28.668 --> 00:02:31.765
times p, so what is this going to be?

00:02:31.765 --> 00:02:33.706
Well, this is going to be equal to,

00:02:33.706 --> 00:02:37.873
we have 500 trials, and then
the probability of success

00:02:39.111 --> 00:02:41.778
on each of these trials is 0.02,

00:02:43.764 --> 00:02:45.681
so it's 500 times 0.02,

00:02:47.014 --> 00:02:48.642
and what is this going to be?

00:02:48.642 --> 00:02:51.809
500 times 2 hundredths is going to be,

00:02:53.559 --> 00:02:55.976
it's going to be equal to 10.

00:02:56.961 --> 00:02:59.846
That is your expected value of the number

00:02:59.846 --> 00:03:02.763
of defective processing
chips, or the mean.

00:03:02.763 --> 00:03:05.939
Now, what about the standard deviation?

00:03:05.939 --> 00:03:09.939
The standard deviation
of our random variable x,

00:03:11.396 --> 00:03:12.961
well, that's just going to be equal to

00:03:12.961 --> 00:03:16.245
the square root of the variance

00:03:16.245 --> 00:03:20.155
of our random variable x,
so I could just write it,

00:03:20.155 --> 00:03:21.899
I'm just writing it all the different ways

00:03:21.899 --> 00:03:23.261
that you might see it because, sometimes,

00:03:23.261 --> 00:03:26.383
the notation is the most
confusing part in statistics.

00:03:26.383 --> 00:03:29.146
So this is going to be
the square root of what?

00:03:29.146 --> 00:03:32.099
Well, the variance of a binomial variable

00:03:32.099 --> 00:03:35.851
is going to be equal
to the number of trials

00:03:35.851 --> 00:03:39.793
times the probability of
success in each trial,

00:03:39.793 --> 00:03:44.597
times one minus the probability
of success in each trial.

00:03:44.597 --> 00:03:49.209
So in this context, this
is going to be equal to,

00:03:49.209 --> 00:03:51.789
you're gonna have the 500,

00:03:51.789 --> 00:03:53.122
500, times 0.02,

00:03:55.925 --> 00:03:58.842
0.02, times one minus 0.02, is .98,

00:04:03.411 --> 00:04:04.578
so times 0.98.

00:04:06.124 --> 00:04:08.319
All of this is under the radical sign.

00:04:08.319 --> 00:04:10.925
I didn't make that
radical sign long enough.

00:04:10.925 --> 00:04:12.452
So what is this going to be?

00:04:12.452 --> 00:04:14.434
Well, let's see, 500 times 0.02,

00:04:14.434 --> 00:04:16.507
we already said that
this is going to be 10.

00:04:16.507 --> 00:04:20.672
10 times 0.98, this is
going to be equal to

00:04:20.672 --> 00:04:22.589
the square root of 9.8,

00:04:25.426 --> 00:04:28.778
so it's going to be, I don't
know. 3-point-something.

00:04:28.778 --> 00:04:30.912
If we want, we can get a calculator out

00:04:30.912 --> 00:04:34.097
to feel a little bit
better about this value.

00:04:34.097 --> 00:04:35.680
I'm gonna take 9.8,

00:04:36.998 --> 00:04:40.795
and then take the square root of it,

00:04:40.795 --> 00:04:45.317
and I get, if I round to
the nearest hundredth, 3.13,

00:04:45.317 --> 00:04:49.484
so this is approximately 3.13
for the standard deviation.

00:04:50.863 --> 00:04:55.028
If I wanted the variance, it would be 9.8,

00:04:55.028 --> 00:04:56.420
but they ask for the standard deviation,

00:04:56.420 --> 00:04:58.351
so that's why we got that.

00:04:58.351 --> 00:05:01.600
All right, hopefully, you enjoyed that.


Expected value of a binomial variable

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.254
- [Tutor] So I've got
a binomial variable X

00:00:02.254 --> 00:00:05.320
and I'm gonna describe
it in very general terms,

00:00:05.320 --> 00:00:07.737
it is the number of successes

00:00:10.690 --> 00:00:11.940
after n trials,

00:00:13.311 --> 00:00:14.561
after n trials,

00:00:19.248 --> 00:00:21.998
where the probability of success,

00:00:24.933 --> 00:00:25.766
success

00:00:27.524 --> 00:00:28.691
for each trial

00:00:31.863 --> 00:00:34.613
is P and this is a reasonable way

00:00:36.832 --> 00:00:40.366
to describe really any
random, any binomial variable,

00:00:40.366 --> 00:00:43.287
we're assuming that each of
these trials are independent,

00:00:43.287 --> 00:00:45.110
the probability stays constant,

00:00:45.110 --> 00:00:48.697
we have a finite number
of trials right over here,

00:00:48.697 --> 00:00:50.388
each trial results in either

00:00:50.388 --> 00:00:52.365
a very clear success or a failure.

00:00:52.365 --> 00:00:54.095
So what we're gonna focus on in this video

00:00:54.095 --> 00:00:57.117
is well, what would be the expected value

00:00:57.117 --> 00:00:59.117
of this binomial variable?

00:00:59.117 --> 00:01:01.617
What would the expected value,

00:01:02.955 --> 00:01:04.538
expected value of X

00:01:06.164 --> 00:01:07.235
be equal to?

00:01:07.235 --> 00:01:10.383
And I will just cut to the
chase and tell you the answer

00:01:10.383 --> 00:01:13.555
and then later in this video,
we'll prove it to ourselves

00:01:13.555 --> 00:01:15.546
a little bit more mathematically.

00:01:15.546 --> 00:01:18.033
The expected value of X, it turns out,

00:01:18.033 --> 00:01:21.361
is just going to be equal
to the number of trials

00:01:21.361 --> 00:01:25.438
times the probability of
success for each of those trials

00:01:25.438 --> 00:01:26.787
and so if you wanted to make

00:01:26.787 --> 00:01:28.643
that a little bit more concrete,

00:01:28.643 --> 00:01:30.143
imagine if a trial

00:01:32.119 --> 00:01:34.249
is a Free Throw,

00:01:34.249 --> 00:01:36.566
taking a shot from the Free Throw line,

00:01:36.566 --> 00:01:37.899
success, success

00:01:40.246 --> 00:01:43.143
is made shot, so you
actually make the shot,

00:01:43.143 --> 00:01:46.019
the ball went in the basket,

00:01:46.019 --> 00:01:49.777
your probability is,
use this yellow color,

00:01:49.777 --> 00:01:52.297
your probability, this would
be your Free Throw percentage,

00:01:52.297 --> 00:01:55.047
so let's say it's 30% or 0.3

00:01:56.694 --> 00:01:58.334
and let's say for the sake of argument,

00:01:58.334 --> 00:02:02.242
that we're taking 10 Free
Throws, so n is equal to 10,

00:02:02.242 --> 00:02:04.698
so this is making it
all a lot more concrete,

00:02:04.698 --> 00:02:08.714
so in this particular
scenario, your expected value,

00:02:08.714 --> 00:02:12.881
your expected value, if X is
the number of made Free Throws,

00:02:13.736 --> 00:02:15.747
after taking 10 Free Throws

00:02:15.747 --> 00:02:18.545
with a Free Throw percentage of 30%,

00:02:18.545 --> 00:02:21.307
well, based on what I just told
you, it would be n times p,

00:02:21.307 --> 00:02:23.873
it would be the number of trials

00:02:23.873 --> 00:02:27.045
times the probability of success
in any one of those trials,

00:02:27.045 --> 00:02:30.727
times 0.3, which is just going to be,

00:02:30.727 --> 00:02:32.557
of course equal to three.

00:02:32.557 --> 00:02:35.246
Now does that make intuitive sense?

00:02:35.246 --> 00:02:36.956
Well, if you're taking 10 shots

00:02:36.956 --> 00:02:38.587
with a 30% Free Throw percentage,

00:02:38.587 --> 00:02:39.780
it actually does feel natural

00:02:39.780 --> 00:02:43.086
that I would expect to make three shots.

00:02:43.086 --> 00:02:44.741
Now with that out of the way,

00:02:44.741 --> 00:02:47.626
let's make ourselves feel
good about this mathematically

00:02:47.626 --> 00:02:49.725
and we're gonna leverage some

00:02:49.725 --> 00:02:51.847
of our expected value properties,

00:02:51.847 --> 00:02:54.044
in particular, we're
gonna leverage the fact

00:02:54.044 --> 00:02:56.356
that if I have the expected value

00:02:56.356 --> 00:02:59.664
of the sum of two
independent random variables,

00:02:59.664 --> 00:03:03.083
let's say X plus Y, it's going to be equal

00:03:03.083 --> 00:03:07.474
to the expected value of X
plus the expected value of Y,

00:03:07.474 --> 00:03:10.636
that we talk about in other videos

00:03:10.636 --> 00:03:13.365
and so assuming this right over here,

00:03:13.365 --> 00:03:16.525
let's construct a new random variable,

00:03:16.525 --> 00:03:19.909
let's call our random variable Y

00:03:19.909 --> 00:03:23.180
and we know the following things about Y,

00:03:23.180 --> 00:03:26.347
the probability that Y is equal to one

00:03:27.623 --> 00:03:28.706
is equal to p

00:03:30.458 --> 00:03:34.778
and the probability
that Y is equal to zero

00:03:34.778 --> 00:03:36.695
is equal to one minus p

00:03:38.045 --> 00:03:41.826
and these are the only two
outcomes for this random variable

00:03:41.826 --> 00:03:45.052
and so you might be seeing
where this is going,

00:03:45.052 --> 00:03:46.736
you could view this random variable,

00:03:46.736 --> 00:03:48.977
it's really representing one trial,

00:03:48.977 --> 00:03:51.289
it becomes one in its success,

00:03:51.289 --> 00:03:53.644
zero when you don't have a success

00:03:53.644 --> 00:03:57.315
and so you could view our
original random variable, X

00:03:57.315 --> 00:04:00.065
right over here as being equal to

00:04:01.687 --> 00:04:05.799
Y plus Y and we're gonna have 10 of these,

00:04:05.799 --> 00:04:07.966
so we're gonna have 10 Ys,

00:04:09.552 --> 00:04:13.369
in the concrete sense, you
could view the random variable Y

00:04:13.369 --> 00:04:15.694
as equaling one, if you make a Free Throw

00:04:15.694 --> 00:04:18.584
and equaling zero, if you
don't make a Free Throw,

00:04:18.584 --> 00:04:21.053
it's really just representing
one of those trials

00:04:21.053 --> 00:04:24.202
and you can view X as the
sum of n of those trials,

00:04:24.202 --> 00:04:26.706
well now actually, let
me be very clear here,

00:04:26.706 --> 00:04:28.366
I immediately went to the concrete,

00:04:28.366 --> 00:04:30.395
but I really should be saying n Ys,

00:04:30.395 --> 00:04:32.827
'cause I wanna stay
general right over here,

00:04:32.827 --> 00:04:34.077
so there are n,

00:04:36.401 --> 00:04:37.328
n Ys right over here,

00:04:37.328 --> 00:04:40.163
this was just a particular example,

00:04:40.163 --> 00:04:42.710
but I am going to try to stay general

00:04:42.710 --> 00:04:43.845
for the rest of the video,

00:04:43.845 --> 00:04:45.407
because now we are really trying

00:04:45.407 --> 00:04:48.278
to prove this result right over here,

00:04:48.278 --> 00:04:52.486
so let's just take the
expected value of both sides,

00:04:52.486 --> 00:04:54.227
so what is it going to be?

00:04:54.227 --> 00:04:57.060
So we get the expected value of X,

00:04:59.360 --> 00:05:00.693
of X is equal to

00:05:02.675 --> 00:05:04.808
well, it's the expected
value of all of this thing,

00:05:04.808 --> 00:05:06.457
but by that property right over here

00:05:06.457 --> 00:05:09.587
is going to be the expected value of Y

00:05:09.587 --> 00:05:12.130
plus the expected value of Y plus,

00:05:12.130 --> 00:05:14.149
and we're gonna do this n times,

00:05:14.149 --> 00:05:16.660
plus the expected value of Y

00:05:16.660 --> 00:05:21.092
and we're gonna have n
of these, so we have n

00:05:21.092 --> 00:05:24.802
and so you could rewrite
this as being equal to,

00:05:24.802 --> 00:05:27.370
this is our n right over here,

00:05:27.370 --> 00:05:28.620
this is n times

00:05:30.071 --> 00:05:32.071
the expected value of Y.

00:05:33.959 --> 00:05:36.959
Now what is the expected value of Y?

00:05:38.515 --> 00:05:39.591
Well, this is pretty straightforward,

00:05:39.591 --> 00:05:41.073
we can actually just do it directly,

00:05:41.073 --> 00:05:44.018
the expected value of Y, let
me just write it over here,

00:05:44.018 --> 00:05:46.479
the expected value of Y

00:05:46.479 --> 00:05:47.942
is just the probability-weighted outcome

00:05:47.942 --> 00:05:50.047
and since there's only two
discrete outcomes here,

00:05:50.047 --> 00:05:51.457
it's pretty easy to calculate,

00:05:51.457 --> 00:05:55.707
we have a probability
of p of getting a one,

00:05:55.707 --> 00:05:57.707
so it's p times one plus

00:05:59.546 --> 00:06:03.114
we have a probability of one minus p

00:06:03.114 --> 00:06:04.785
of getting a zero,

00:06:04.785 --> 00:06:06.467
well, what does this simplify to?

00:06:06.467 --> 00:06:07.984
Well, zero times anything, that's zero

00:06:07.984 --> 00:06:11.553
and then you have one times
p, this is just equal to p,

00:06:11.553 --> 00:06:13.776
so expected value of Y is just equal to p

00:06:13.776 --> 00:06:17.113
and so there you have it, we
get the expected value of X

00:06:17.113 --> 00:06:18.809
is 10 times the expected value,

00:06:18.809 --> 00:06:21.132
or the expected value of X is n

00:06:21.132 --> 00:06:22.967
times the expected value of Y

00:06:22.967 --> 00:06:25.203
and the expected value of Y is p,

00:06:25.203 --> 00:06:28.703
so the expected value of X is equal to np,

00:06:29.563 --> 00:06:32.063
hope you feel good about that.


Variance of a binomial variable

WEBVTT
Kind: captions
Language: en

00:00:00.564 --> 00:00:01.636
- [Instructor] What we're
going to do in this video

00:00:01.636 --> 00:00:04.396
is continue our journey
trying to understand

00:00:04.396 --> 00:00:06.923
what the expected value
and what the variance

00:00:06.923 --> 00:00:10.199
of a binomial variable is going to be

00:00:10.199 --> 00:00:12.692
or what the expected value or the variance

00:00:12.692 --> 00:00:14.476
of a binominal distribution is going to be

00:00:14.476 --> 00:00:18.171
which is just the distribution
of a binomial variable.

00:00:18.171 --> 00:00:19.810
And so, like in the last video

00:00:19.810 --> 00:00:22.025
I have this binomial variable X

00:00:22.025 --> 00:00:23.653
that's defined in a very general sense.

00:00:23.653 --> 00:00:26.405
It's the number of
successes from N trials,

00:00:26.405 --> 00:00:27.919
so it's a finite number of trials

00:00:27.919 --> 00:00:30.662
where the probability of
success is equal to P,

00:00:30.662 --> 00:00:33.334
so the probability is
constant across the trials

00:00:33.334 --> 00:00:36.341
for each of these independent trials,

00:00:36.341 --> 00:00:38.266
so the probability of success in one trial

00:00:38.266 --> 00:00:42.262
is not dependent on what
happened in the other rials.

00:00:42.262 --> 00:00:44.371
And we also talked in that previous video

00:00:44.371 --> 00:00:46.311
where we talked about the expected value

00:00:46.311 --> 00:00:48.739
of this binomial variable

00:00:48.739 --> 00:00:50.859
we said hey, it could be viewed

00:00:50.859 --> 00:00:52.452
that this binomial variable

00:00:52.452 --> 00:00:53.857
can be viewed as the sum

00:00:53.857 --> 00:00:56.426
of N of what you could really consider

00:00:56.426 --> 00:00:58.246
to be a Bernoulli variable here.

00:00:58.246 --> 00:01:00.675
So, this variable, this random variable Y,

00:01:00.675 --> 00:01:02.299
the probability that's equal to one,

00:01:02.299 --> 00:01:05.006
you could do that as a
success is equal to P.

00:01:05.006 --> 00:01:06.249
The probability that it's a failure

00:01:06.249 --> 00:01:09.655
that Y is equal to zero is one minus P,

00:01:09.655 --> 00:01:13.410
so you could view Y, the outcome of Y

00:01:13.410 --> 00:01:15.027
or whether Y is one or zero

00:01:15.027 --> 00:01:17.016
is really whether we had a success or not

00:01:17.016 --> 00:01:18.402
in each of these trials,

00:01:18.402 --> 00:01:19.890
so if you add up N Ys,

00:01:19.890 --> 00:01:22.601
then you are going to get X

00:01:22.601 --> 00:01:24.373
and we use that information to figure out

00:01:24.373 --> 00:01:27.245
what the expected value
of X is going to be

00:01:27.245 --> 00:01:28.804
because the expected value of Y

00:01:28.804 --> 00:01:31.956
is pretty straightforward
to directly compute.

00:01:31.956 --> 00:01:33.529
Expected value of Y

00:01:33.529 --> 00:01:35.607
is just the probability weighted outcomes.

00:01:35.607 --> 00:01:37.274
So, it's P times one

00:01:40.166 --> 00:01:41.583
plus one minus P,

00:01:43.090 --> 00:01:46.090
one minus P, times zero, times zero.

00:01:48.306 --> 00:01:50.433
This whole term's gonna be zero

00:01:50.433 --> 00:01:53.380
and so, the expected value
of Y is really just P

00:01:53.380 --> 00:01:55.550
and so, if you said the
expected value of X,

00:01:55.550 --> 00:01:57.038
well, that's just going to be,

00:01:57.038 --> 00:01:58.158
let me just write it over here,

00:01:58.158 --> 00:01:59.064
this is all review,

00:01:59.064 --> 00:02:02.481
we could say that the expected value of X

00:02:04.631 --> 00:02:06.595
is just going to be equal to,

00:02:06.595 --> 00:02:08.392
we know from our expected value properties

00:02:08.392 --> 00:02:11.084
that it's going to be equal to the sum

00:02:11.084 --> 00:02:13.394
of the expected values of these N Ys,

00:02:13.394 --> 00:02:15.394
or you could say it is N

00:02:16.301 --> 00:02:19.010
times the expected value,

00:02:19.010 --> 00:02:21.264
times the expected value of Y,

00:02:21.264 --> 00:02:23.345
the expected value of Y is P,

00:02:23.345 --> 00:02:26.845
so this is going to be equal to N times P.

00:02:28.394 --> 00:02:30.917
Now, we're gonna do the same idea

00:02:30.917 --> 00:02:33.005
to figure out what the variance of X

00:02:33.005 --> 00:02:34.833
is going to be equal to

00:02:34.833 --> 00:02:37.771
because we could see, we know
from our variance properties,

00:02:37.771 --> 00:02:38.979
you can't do this with standard deviation

00:02:38.979 --> 00:02:40.604
but you could do it with variance

00:02:40.604 --> 00:02:41.964
and then once you figure out the variance,

00:02:41.964 --> 00:02:42.797
you just take the square root

00:02:42.797 --> 00:02:44.420
for the standard deviation,

00:02:44.420 --> 00:02:47.394
the variance of X is similarly

00:02:47.394 --> 00:02:51.352
going to be the sum of the
variances of these N Ys.

00:02:51.352 --> 00:02:54.310
So, it's gonna be similarly N

00:02:54.310 --> 00:02:58.143
times the variance, N
times the variance of Y.

00:02:59.899 --> 00:03:04.722
So, this all boils down to
what is the variance of Y

00:03:04.722 --> 00:03:06.083
going to be equal be?

00:03:06.083 --> 00:03:07.770
So, let me scroll over a little bit,

00:03:07.770 --> 00:03:09.417
get a little bit of more real estate

00:03:09.417 --> 00:03:13.000
and I will figure that
out right over here.

00:03:14.667 --> 00:03:19.350
Alright, so we wanna figure
out the variance of Y,

00:03:19.350 --> 00:03:20.683
so variance of Y

00:03:22.302 --> 00:03:24.074
is going to be equal to what?

00:03:24.074 --> 00:03:26.247
Well, here it's going
to be the probability

00:03:26.247 --> 00:03:30.405
squared distances from the expected value.

00:03:30.405 --> 00:03:32.905
So, we have a probability of P

00:03:35.267 --> 00:03:38.236
where what is going to
be our squared distance

00:03:38.236 --> 00:03:40.116
from the expected value?

00:03:40.116 --> 00:03:43.295
Well, we're going to get a
one with a probability of P,

00:03:43.295 --> 00:03:45.383
so in that case our distance from the mean

00:03:45.383 --> 00:03:48.148
or from the expected value, we're at one,

00:03:48.148 --> 00:03:49.603
the expected value we already know

00:03:49.603 --> 00:03:50.770
is equal to P,

00:03:52.672 --> 00:03:55.124
so that's that for that possible outcome,

00:03:55.124 --> 00:03:57.868
the squared distance times
its probability weight

00:03:57.868 --> 00:03:59.356
and then we have,

00:03:59.356 --> 00:04:00.867
actually let me scroll over,

00:04:00.867 --> 00:04:01.989
well, I'll just do it right over here,

00:04:01.989 --> 00:04:05.489
plus we have a probability of one minus P,

00:04:07.807 --> 00:04:10.908
one minus P for the
other possible outcome,

00:04:10.908 --> 00:04:11.741
so in that outcome

00:04:11.741 --> 00:04:12.908
we are at zero

00:04:13.754 --> 00:04:16.877
and the difference between
zero and our expected value?

00:04:16.877 --> 00:04:19.383
Well, that's just going to be zero minus P

00:04:19.383 --> 00:04:23.550
and once again we are going
to square that quantity

00:04:25.284 --> 00:04:27.921
and so, this is the expression
for the variance of Y

00:04:27.921 --> 00:04:29.540
and we can simplify it a little bit.

00:04:29.540 --> 00:04:32.496
So, this is all going to be equal to,

00:04:32.496 --> 00:04:35.079
so, P times one minus P squared

00:04:36.620 --> 00:04:38.657
and then is just going to be P squared

00:04:38.657 --> 00:04:40.138
times one minus P

00:04:40.138 --> 00:04:42.805
plus P squared times one minus P

00:04:44.264 --> 00:04:48.933
and let's see, we can factor
out a P times one minus P,

00:04:48.933 --> 00:04:50.352
so what is that going to be left with?

00:04:50.352 --> 00:04:52.686
So, if we factor out a P
times one minus P here,

00:04:52.686 --> 00:04:55.569
we're just going to be
left with a one minus P

00:04:55.569 --> 00:04:59.586
and if we factor out a P
times one minus P here,

00:04:59.586 --> 00:05:02.686
we're just going to have a plus P.

00:05:02.686 --> 00:05:04.453
These two cancel out.

00:05:04.453 --> 00:05:06.480
This is just this whole
thing is just a one.

00:05:06.480 --> 00:05:09.815
So, you're left with P times one minus P

00:05:09.815 --> 00:05:12.140
which is indeed the variance
for a binomial variable.

00:05:12.140 --> 00:05:13.346
We actually proved that in other videos.

00:05:13.346 --> 00:05:15.144
I guess it doesn't hurt to see it again

00:05:15.144 --> 00:05:15.977
but there you have.

00:05:15.977 --> 00:05:18.580
We know what the variance of Y is.

00:05:18.580 --> 00:05:20.993
It is P times one minus P

00:05:20.993 --> 00:05:24.494
and the variance of X is just
N times the variance of Y,

00:05:24.494 --> 00:05:27.213
so there we go, we deserve
a little bit of a drum roll,

00:05:27.213 --> 00:05:30.616
the variance of X is equal to N

00:05:30.616 --> 00:05:32.783
times P times one minus P.

00:05:34.947 --> 00:05:36.661
So, if we were to take
the concrete example

00:05:36.661 --> 00:05:38.209
of the last video

00:05:38.209 --> 00:05:40.831
where if I were to take 10 free throws,

00:05:40.831 --> 00:05:44.068
so each trial is a shot, is a free throw,

00:05:44.068 --> 00:05:45.796
so if I were to take 10 free throws

00:05:45.796 --> 00:05:49.676
and my probability of success is 0.3,

00:05:49.676 --> 00:05:52.107
I have a 30% free throw percentage,

00:05:52.107 --> 00:05:54.536
the variance that I would expect to see,

00:05:54.536 --> 00:05:56.869
so in that case the variance

00:05:58.344 --> 00:06:00.425
if X is the number of free throws

00:06:00.425 --> 00:06:03.177
I make after these 10 shots,

00:06:03.177 --> 00:06:05.010
my variance will be 10

00:06:06.818 --> 00:06:07.651
times 0.3,

00:06:10.909 --> 00:06:12.909
0.3 times one minus 0.3,

00:06:14.422 --> 00:06:15.589
so 0.7 and so,

00:06:18.215 --> 00:06:19.676
that would be what?

00:06:19.676 --> 00:06:23.982
This right over, so this
would be equal to 10

00:06:23.982 --> 00:06:26.399
times .3 times .7 times 0.21,

00:06:29.434 --> 00:06:32.711
so my variance in this situation

00:06:32.711 --> 00:06:35.044
is going to be equal to 2.1.

00:06:36.712 --> 00:06:37.962
Is equal to 2.1

00:06:39.637 --> 00:06:42.122
and if I wanted to figure
out the standard deviation

00:06:42.122 --> 00:06:43.349
of this right over here,

00:06:43.349 --> 00:06:45.152
I would just take the square root of this,

00:06:45.152 --> 00:06:46.273
so if we want the standard deviation,

00:06:46.273 --> 00:06:47.331
just take the square root

00:06:47.331 --> 00:06:49.898
of this expression right over here.


Binomial variables

WEBVTT
Kind: captions
Language: en

00:00:00.291 --> 00:00:01.571
- What we're going to do in this video

00:00:01.571 --> 00:00:05.341
is talk about a special
class of random variables

00:00:05.341 --> 00:00:07.674
known as binomial variables.

00:00:09.533 --> 00:00:12.882
And as we will see as we build
up our understanding of them,

00:00:12.882 --> 00:00:15.085
not only are they interesting
in their own right,

00:00:15.085 --> 00:00:16.675
but there's a lot of very powerful

00:00:16.675 --> 00:00:18.359
probability and statistics

00:00:18.359 --> 00:00:21.035
that we can do based on our understanding

00:00:21.035 --> 00:00:23.288
of binomial variables.

00:00:23.288 --> 00:00:26.123
So to make things concrete
as quickly as possible,

00:00:26.123 --> 00:00:28.592
I'll start with a very tangible example

00:00:28.592 --> 00:00:29.734
of a binomial variable

00:00:29.734 --> 00:00:31.434
and then we'll think a
little bit more abstractly

00:00:31.434 --> 00:00:34.097
abut what makes it binomial.

00:00:34.097 --> 00:00:36.089
So let's say that I have a coin.

00:00:36.089 --> 00:00:38.106
This is my coin here.

00:00:38.106 --> 00:00:40.922
It doesn't even have to be a fair coin.

00:00:40.922 --> 00:00:42.410
Let me just draw this really fast.

00:00:42.410 --> 00:00:43.478
So that's my coin.

00:00:43.478 --> 00:00:46.224
And let's say on a
given flip of that coin,

00:00:46.224 --> 00:00:50.504
the probability that I get
heads is zero point six

00:00:50.504 --> 00:00:52.604
and the probability that I'd get tails,

00:00:52.604 --> 00:00:54.496
well it'd be one minus zero point six

00:00:54.496 --> 00:00:56.079
or zero point four.

00:00:58.453 --> 00:01:00.911
And what I'm going to do
is I'm going to define

00:01:00.911 --> 00:01:03.277
a random variable X

00:01:03.277 --> 00:01:06.058
as being equal to the number

00:01:06.058 --> 00:01:06.891
of heads

00:01:09.314 --> 00:01:10.147
after

00:01:11.958 --> 00:01:12.791
after

00:01:13.881 --> 00:01:15.038
ten

00:01:15.038 --> 00:01:15.871
flips

00:01:17.161 --> 00:01:18.078
of my coin.

00:01:20.319 --> 00:01:23.736
Now, what makes this a binomial variable?

00:01:24.814 --> 00:01:26.289
Well, one of the first conditions

00:01:26.289 --> 00:01:28.845
that's often given for a binomial variable

00:01:28.845 --> 00:01:31.324
is that it's made up of a finite number

00:01:31.324 --> 00:01:33.930
of independent trials.

00:01:33.930 --> 00:01:34.928
So,

00:01:34.928 --> 00:01:35.928
it's made up

00:01:36.934 --> 00:01:37.767
made up of

00:01:39.740 --> 00:01:40.657
independent

00:01:43.471 --> 00:01:45.862
independent trials.

00:01:45.862 --> 00:01:49.535
Now, what do I mean by independent trials?

00:01:49.535 --> 00:01:52.505
Well, a trial is each flip of my coin.

00:01:52.505 --> 00:01:53.338
So a flip

00:01:54.548 --> 00:01:56.741
is equal to a trial

00:01:56.741 --> 00:01:59.694
in the language of this
statement that I just made.

00:01:59.694 --> 00:02:02.836
And what do I mean by each flip

00:02:02.836 --> 00:02:06.233
or each trial being independent?

00:02:06.233 --> 00:02:09.123
Well the probability of
whether I get heads or tails

00:02:09.123 --> 00:02:10.904
on each flip

00:02:10.904 --> 00:02:14.322
are independent of whether
I just got heads or tails

00:02:14.322 --> 00:02:16.201
on some previous flip.

00:02:16.201 --> 00:02:17.977
So, in this case, we are made up of

00:02:17.977 --> 00:02:19.560
independent trials.

00:02:20.514 --> 00:02:22.114
Now, another condition is

00:02:22.114 --> 00:02:25.179
each trial can be clearly classified

00:02:25.179 --> 00:02:27.190
as either a success or failure.

00:02:27.190 --> 00:02:28.460
Or another way of thinking about it:

00:02:28.460 --> 00:02:33.386
Each trial clearly has one
of two discreet outcomes.

00:02:33.386 --> 00:02:34.553
So each trial,

00:02:36.003 --> 00:02:37.337
and the example I'm giving,

00:02:37.337 --> 00:02:39.254
the flip is a trial,

00:02:39.254 --> 00:02:40.671
can be classified

00:02:44.197 --> 00:02:45.030
classified

00:02:46.089 --> 00:02:47.715
as

00:02:47.715 --> 00:02:48.548
either

00:02:49.632 --> 00:02:50.465
success

00:02:52.365 --> 00:02:53.282
or failure.

00:02:55.139 --> 00:02:57.867
So, in the context of
this random variable X,

00:02:57.867 --> 00:03:00.416
we could define heads as a success

00:03:00.416 --> 00:03:03.564
because that's what we
are happening to count up.

00:03:03.564 --> 00:03:06.267
And so you're either going
to have success or failure.

00:03:06.267 --> 00:03:09.007
You're either going to have heads or tails

00:03:09.007 --> 00:03:11.686
on each of these trials.

00:03:11.686 --> 00:03:15.111
Now another condition for
being a binomial variable

00:03:15.111 --> 00:03:17.902
is that you have a fixed number of trials.

00:03:17.902 --> 00:03:19.036
Fixed

00:03:19.036 --> 00:03:20.312
number

00:03:20.312 --> 00:03:21.712
of

00:03:21.712 --> 00:03:22.889
trials.

00:03:22.889 --> 00:03:23.911
So in this case,

00:03:23.911 --> 00:03:26.555
we're saying that we have ten trials,

00:03:26.555 --> 00:03:28.388
ten flips of our coin.

00:03:29.533 --> 00:03:33.366
And then the last condition
is the probability

00:03:36.129 --> 00:03:37.046
of success,

00:03:38.654 --> 00:03:40.779
in this context success is a heads,

00:03:40.779 --> 00:03:41.946
on each trial,

00:03:44.482 --> 00:03:45.399
each trial,

00:03:46.362 --> 00:03:47.362
is constant.

00:03:49.224 --> 00:03:50.163
And we've already talked about it.

00:03:50.163 --> 00:03:51.948
On each trial on each flip,

00:03:51.948 --> 00:03:53.061
the probability of heads

00:03:53.061 --> 00:03:55.958
is going to stay at zero point six.

00:03:55.958 --> 00:03:57.756
If for some reason that were to change

00:03:57.756 --> 00:03:59.434
from trial to trial,

00:03:59.434 --> 00:04:01.277
maybe if you were to swap the coin

00:04:01.277 --> 00:04:03.803
and each coin had a different probability

00:04:03.803 --> 00:04:08.006
then this would no longer
be a binomial variable.

00:04:08.006 --> 00:04:08.839
And so you might say,

00:04:08.839 --> 00:04:09.761
"Okay, that's reasonable,

00:04:09.761 --> 00:04:12.477
I get why this is a binomial variable.

00:04:12.477 --> 00:04:13.310
Can you give me an example

00:04:13.310 --> 00:04:16.157
of something that is not
a binomial variable?"

00:04:16.157 --> 00:04:20.083
Well let's say that I were
to define the variable Y

00:04:20.083 --> 00:04:23.166
and it's equal to the number of kings

00:04:27.197 --> 00:04:28.030
after

00:04:29.355 --> 00:04:30.188
taking

00:04:31.648 --> 00:04:32.481
two cards

00:04:34.072 --> 00:04:35.443
from

00:04:35.443 --> 00:04:38.126
a standard deck of cards.

00:04:38.126 --> 00:04:39.293
Standard deck.

00:04:40.737 --> 00:04:42.377
Without replacement.

00:04:42.377 --> 00:04:44.044
Without replacement.

00:04:46.392 --> 00:04:47.395
So you might immediately say,

00:04:47.395 --> 00:04:50.645
"Well, this feels like
it could be binomial."

00:04:50.645 --> 00:04:53.472
We have...each trial can be classified

00:04:53.472 --> 00:04:55.342
as either a success or failure.

00:04:55.342 --> 00:04:58.030
Each trial is when I take a card out.

00:04:58.030 --> 00:05:00.937
If I get a king that looks
like that would be a success.

00:05:00.937 --> 00:05:03.650
If I don't get a king
that would be a failure.

00:05:03.650 --> 00:05:06.619
So it seems to meet that right over there.

00:05:06.619 --> 00:05:08.524
It has a fixed number of trials.

00:05:08.524 --> 00:05:10.533
I'm taking two cards out of the deck

00:05:10.533 --> 00:05:12.951
so it seems to meet that.

00:05:12.951 --> 00:05:14.586
But what about these conditions

00:05:14.586 --> 00:05:16.778
that it's made up of independent trials

00:05:16.778 --> 00:05:19.147
or that the probability
of success on each trial

00:05:19.147 --> 00:05:20.147
is constant?

00:05:21.059 --> 00:05:22.102
Well,

00:05:22.102 --> 00:05:23.335
if I get a king

00:05:23.335 --> 00:05:25.786
the probability of king
on the first trial,

00:05:25.786 --> 00:05:27.647
probability I say king

00:05:27.647 --> 00:05:28.814
on first trial

00:05:31.037 --> 00:05:32.703
would be equal to, well,

00:05:32.703 --> 00:05:34.909
out of a deck of 52 cards,

00:05:34.909 --> 00:05:37.438
you're going to have four kings in it.

00:05:37.438 --> 00:05:40.300
So the probability of a
king on the first trial

00:05:40.300 --> 00:05:42.789
would be four out of 52.

00:05:42.789 --> 00:05:45.646
But what about the
probability of getting a king

00:05:45.646 --> 00:05:46.729
on the second

00:05:47.932 --> 00:05:50.053
on the second trial?

00:05:50.053 --> 00:05:51.907
What would this be equal to?

00:05:51.907 --> 00:05:55.969
Well, it depends on what
happened on the first trial.

00:05:55.969 --> 00:05:58.710
If the first trial you had a king,

00:05:58.710 --> 00:05:59.850
well then you would have,

00:05:59.850 --> 00:06:01.735
so let's see, this would be the situation

00:06:01.735 --> 00:06:03.824
given first trial,

00:06:03.824 --> 00:06:05.518
first king,

00:06:05.518 --> 00:06:08.375
well now there would be three kings left

00:06:08.375 --> 00:06:10.208
in a deck of 51 cards.

00:06:12.154 --> 00:06:14.419
But if you did not get a
king on the first trial,

00:06:14.419 --> 00:06:18.769
now you have four kings
in a deck of 51 cards

00:06:18.769 --> 00:06:20.666
because, remember, we're
doing it without replacement.

00:06:20.666 --> 00:06:22.933
You're just taking that
first card, whatever you did,

00:06:22.933 --> 00:06:25.424
and you're taking it aside.

00:06:25.424 --> 00:06:27.090
So what's interesting here is

00:06:27.090 --> 00:06:29.590
this is not made up of independent trials.

00:06:29.590 --> 00:06:31.426
It does not meet this condition.

00:06:31.426 --> 00:06:33.338
The probability on your second trial

00:06:33.338 --> 00:06:37.194
is dependent on what
happens on your first trial.

00:06:37.194 --> 00:06:38.279
And another way to think about it

00:06:38.279 --> 00:06:41.430
is because we aren't replacing
each card that we're picking,

00:06:41.430 --> 00:06:43.399
the probability of success on each trial

00:06:43.399 --> 00:06:45.824
also is not constant.

00:06:45.824 --> 00:06:48.594
And so that's why this right over here

00:06:48.594 --> 00:06:50.844
is not a binomial variable.

00:06:52.204 --> 00:06:53.650
Now, if Y,

00:06:53.650 --> 00:06:55.564
if we got rid of

00:06:55.564 --> 00:06:56.868
without replacement

00:06:56.868 --> 00:07:00.818
and if we said we did replace
every card after we picked it

00:07:00.818 --> 00:07:02.905
then things would be different.

00:07:02.905 --> 00:07:06.228
Then we actually would be
looking at a binomial variable.

00:07:06.228 --> 00:07:08.050
So instead of without replacement

00:07:08.050 --> 00:07:10.736
if I just said with replacement,

00:07:10.736 --> 00:07:14.216
well then your probability
of a king on each trial

00:07:14.216 --> 00:07:17.271
is going to be four out of 52.

00:07:17.271 --> 00:07:19.427
You have a finite number of trials.

00:07:19.427 --> 00:07:22.490
You're probability of success
is going to stay constant

00:07:22.490 --> 00:07:24.971
and they would be independent.

00:07:24.971 --> 00:07:27.664
And obviously each trial
could easily be classified

00:07:27.664 --> 00:07:30.393
as either a success or a failure.