Shorting stock 2

WEBVTT
Kind: captions
Language: en

00:00:01.750 --> 00:00:04.250
Let's review what we went over
in the last video, and one of

00:00:04.250 --> 00:00:06.010
you all actually commented that
it would be a good idea

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to draw a timeline.

00:00:06.950 --> 00:00:09.780
So I'll draw a timeline.

00:00:09.780 --> 00:00:10.080
Short.

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So we're learning about
short selling.

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And in the last example-- let
me do the timeline where

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things work out well for
the short seller.

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So let me draw the stock
price of IBM.

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Let me make this its--
OK, here we go.

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So let's say that this is-- that
could be our timeline,

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it's by day.

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Let me draw the stock of IBM, it
could look something like--

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that's my y-axis.

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Let's say the stock right now
is at $100, it's trading

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someplace like that.

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And let's say it does
that later, right?

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But we're sitting at this
point right here-- we're

00:00:48.910 --> 00:00:51.980
sitting at, let's
call this day 0.

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So what does the short
seller do?

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So let's say the short seller,
right now-- let me see if I

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can draw his balance sheet.

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So right now, the short seller,
he has assets and

00:01:04.959 --> 00:01:06.209
liabilities.

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His assets-- I won't worry about
collateral requirements

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and all of that right now.

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But usually he already has to
have some assets ahead of time

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for him to be able
to borrow shares.

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But, actually-- let
me give him some

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collateral ahead of time.

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So let's say that he already
has $60 in his account.

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He has $60 of assets on day 0.

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And then this is the day that
he says, you know what?

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I've done my analysis and I
think IBM-- he doesn't see

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this part of the stock price,
I mean, it would

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be great if he did.

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Then you could short
with conviction.

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But all he sees is
the past, right?

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If he did a stock chart he would
just see-- let me switch

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colors-- he would just see this
green part right here.

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He wouldn't see all the stuff
that's in the future.

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But he has a lot of conviction
that IBM is going to go down.

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So what he does is, he borrows
a share of IBM on that day.

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So then on this day he
borrows one share.

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So he has-- let's call
that IBM-- one IBM.

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And he also owes one IBM.

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Right?

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Right after you borrow it,
before you do anything into

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it, you have it as an asset,
and you also owe it back.

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And if you wanted unwind the
borrowing of it, you could

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just give it back.

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But what he does at this point
is he sells this IBM.

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He sells that share and
he gets $100 for it.

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Because that was just
the market price.

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That's what people were willing
to trade IBM shares

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for at that point in time.

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That's day 0.

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Then let's say IBM reports
its earnings, and

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they're really bad.

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And that happened on, I
don't know, probably

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happened on this day.

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IBM reports.

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And the stock tends to
go down, down, down.

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People take a long time
to realize how

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bad the report was.

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And here at this day, once the
stock has reached $50, our

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short seller says OK,
that's enough.

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I don't think the stock's going
to drop a lot more.

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So on day-- let's call
this day 10.

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10 days have gone by.

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Day 10, he decides to cover.

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So going into day 10, this
is his balance sheet--

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let me redraw it.

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So going in to day 10,
what does he have?

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He has $160.

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The $60 he had before, just
by actually working.

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And he owes-- this is his asset,
and his liabilities is

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he owes one share of
IBM to the broker.

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And the broker really owes it to
one of the shareholders of

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IBM who happened to be keeping
the share with the broker.

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And he wants to cover.

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So what he does is, he takes
$100-- no, no sorry, he

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doesn't take $100.

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Now shares of IBM only
cost $50, right?

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So he takes $50 to buy a
share, to buy one IBM.

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So instead of $160, he
now has $110 and he

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has a share of IBM.

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And then what he does is, he
takes this share of IBM and

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then gives it to the brokerage
to pay off his liability.

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So then he's done.

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He's left with no liabilities,
and just $110 of assets.

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So he made $50.

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So hopefully that clarifies it
up a little bit, in that he

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sold here, and bought here.

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It's the reverse of a lot of
stock, it's almost like you're

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acting in reverse time.

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But this was a very good
scenario for the short seller.

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But he very easily could
have made a blunder.

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Let's see what could have been
a blunderous scenario.

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Let me draw a different
stock chart for IBM.

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So let me draw the stock up to
the day in question, and we

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said it was looking something
like this, where it was

00:04:57.570 --> 00:04:59.640
trading right at around $100.

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And this is the day that our
short seller decides to short

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it, and this happens.

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Right?

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He essentially borrows
a share of IBM.

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So he has a one IBM
share liability.

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He sells that share and
he collects $100.

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And then let's say IBM reports
on this day, so this is day 0.

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Now IBM reports, and it's
actually great.

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They did way better than anyone
could have expected.

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So then the IBM shares
skyrocket, and

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they go to this level.

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And at this point this-- and
I'll talk more about short

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psychology and short squeezing,
and all that-- but

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maybe here he's like, oh no,
this is just a temporary blip,

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let me keep holding
my position.

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But then the stock keeps going
up and up and he says oh, this

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is just temporary, it's
going to go back down.

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But at some point, his tolerance
for pain has been

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tapped out.

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And let's say IBM
gets to $150.

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He says, I can't handle
this anymore.

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And I think you're already
noticing a very negative

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dynamic or a highly risky
dynamic that occurs with

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shorts, is that you can lose an
arbitrary amount of money.

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Because what's happening now?

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Let's say he wants to
cover it right now.

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This is day 10 in this
alternate universe.

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So now, what are his assets
and his liabilities?

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Going in to day 10, his asset,
we said, was $160.

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Because he had short
sold, he had $160.

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But he owes one share of IBM.

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For him to unwind this, to pay
back the share of IBM, what

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does he have to do?

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He has to go out into the market
and buy a share of IBM

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at this higher price, at $150.

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So when he goes out, instead of
$160 he has to use 100-- so

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he has $10-- and then he uses
$150 of that to go buy-- $150

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of the $160 to buy
a share of IBM.

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So then he gets a
share of IBM.

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And then he can pay that share
back to the broker and cancel

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out his position.

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And he's left with just $10.

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So in this scenario when
the stock price rose by

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$50, he lost $50.

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So he sold low and then
he bought high, right?

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And the really risky thing that
maybe is apparent to you

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now about short selling
is that his loss

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could have been infinite.

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What if IBM, instead of going to
$150, what if went to $200?

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Then he would have lost $200--
if it went to $200, he would

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have lost $100.

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If it went to $300, he
would have lost $200.

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So his loss isn't just
the amount of the

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original short position.

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It isn't just the $100 or
whatever the original

00:07:40.620 --> 00:07:41.730
price of IBM was.

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It can be an infinite amount,
so it really can kill your

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balance sheet.

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Or really make you broke.

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While when you go in the long
side of things, if I were to

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buy IBM here.

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Let's say I'm the guy that
bought this share from the

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short seller.

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What's my worst case scenario?

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Well the worst thing I can have
happen is that the share

00:08:00.740 --> 00:08:01.890
of IBM goes to 0.

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So my loss is really,
I can just go to 0.

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I won't end up owing someone an
infinite amount of money.

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So short selling, inherently,
because of this infinite, you

00:08:12.870 --> 00:08:14.880
could say downside to the
short seller, right?

00:08:14.880 --> 00:08:17.070
They can lose an infinite
amount of money.

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They have to be really careful
about how they make their

00:08:19.370 --> 00:08:22.760
positions and how they protect
themselves from this

00:08:22.760 --> 00:08:23.740
eventuality.

00:08:23.740 --> 00:08:25.610
And we'll talk a little bit
about things like margin

00:08:25.610 --> 00:08:28.100
requirements and things like
that in the future, that

00:08:28.100 --> 00:08:33.260
essentially make sure-- are
the broker's way of making

00:08:33.260 --> 00:08:39.080
sure that the short seller can
actually-- is good to buy back

00:08:39.080 --> 00:08:40.039
the shares.

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Anyway, see you in
the next video.


Shorting stock

WEBVTT
Kind: captions
Language: en

00:00:01.660 --> 00:00:06.530
You've probably heard the term
short sale, and have at least

00:00:06.530 --> 00:00:08.390
a general-- oh, what
did I do with that?

00:00:08.390 --> 00:00:10.440
Oh there it is, I scrolled
down-- you probably have a

00:00:10.440 --> 00:00:13.800
general idea that it means, to
some degree, making a bet that

00:00:13.800 --> 00:00:15.060
a stock will fall.

00:00:15.060 --> 00:00:17.800
And there's been a lot of news
lately that maybe short

00:00:17.800 --> 00:00:20.480
sellers are to blame for a lot
of the stock market crashes,

00:00:20.480 --> 00:00:21.580
or what's happening
with banks.

00:00:21.580 --> 00:00:23.620
So I thought it would be
worthwhile to at least explain

00:00:23.620 --> 00:00:25.040
what a short sale is.

00:00:25.040 --> 00:00:27.210
And then we could talk a little
bit about whether

00:00:27.210 --> 00:00:30.030
they're positive or negative,
or they could be both.

00:00:30.030 --> 00:00:33.650
So you are right if you think
that a short sale is some type

00:00:33.650 --> 00:00:35.600
of a bet that a stock
can go down.

00:00:35.600 --> 00:00:36.920
But how does it work?

00:00:36.920 --> 00:00:39.280
So let's say that, I don't know,
IBM-- and I don't know

00:00:39.280 --> 00:00:44.200
where IBM is trading right now--
but let's say IBM is

00:00:44.200 --> 00:00:49.760
right now-- the last
trade was at $100.

00:00:49.760 --> 00:00:50.680
And I'm convinced.

00:00:50.680 --> 00:00:53.790
I've done my research, I've done
my analysis, and I'm just

00:00:53.790 --> 00:00:57.190
convinced that the stock price
of IBM is going to fall.

00:00:57.190 --> 00:01:00.620
I think IBM is going to have a
horrible quarter next quarter.

00:01:00.620 --> 00:01:02.670
That some other competitor
is going to take

00:01:02.670 --> 00:01:03.570
their business, whatever.

00:01:03.570 --> 00:01:06.690
I'm convinced it's going
to fall to $50.

00:01:06.690 --> 00:01:07.750
This is what I think.

00:01:07.750 --> 00:01:10.390
I think it's going to fall
to $50, based on

00:01:10.390 --> 00:01:13.070
my deep stock analysis.

00:01:13.070 --> 00:01:16.450
And so, the general idea is that
I should-- well I won't

00:01:16.450 --> 00:01:19.270
make any value judgment right
now-- but I want to know if

00:01:19.270 --> 00:01:22.650
there's a way that I can make
money off of my deep analysis.

00:01:22.650 --> 00:01:26.480
And the way I do that is I go
to my brokerage-- so let me

00:01:26.480 --> 00:01:33.510
draw some diagrams. So this is
my brokerage, this is my

00:01:33.510 --> 00:01:39.320
broker, and this is
me right here.

00:01:39.320 --> 00:01:41.530
This is me.

00:01:41.530 --> 00:01:44.310
I have a top hat.

00:01:44.310 --> 00:01:46.990
Anyway, so I go to the broker
and I say broker, I would like

00:01:46.990 --> 00:01:48.840
to short IBM.

00:01:48.840 --> 00:01:51.500
I would like-- let's just say
for simplicity I want to short

00:01:51.500 --> 00:01:52.950
one share of IBM.

00:01:52.950 --> 00:01:55.400
So what I do is I go to the
broker, and I say broker, can

00:01:55.400 --> 00:01:58.220
I borrow a share of IBM?

00:01:58.220 --> 00:02:00.140
And this is a little unintuitive
the first time

00:02:00.140 --> 00:02:00.850
people learn about it.

00:02:00.850 --> 00:02:03.100
It was a little unintuitive
for me the first time I

00:02:03.100 --> 00:02:05.490
learned about the notion
of borrowing a share.

00:02:05.490 --> 00:02:07.480
But you can also view it
as renting a share.

00:02:07.480 --> 00:02:08.539
And you might say well,
where's the

00:02:08.539 --> 00:02:09.800
broker getting it from?

00:02:09.800 --> 00:02:12.790
Well this broker's got
a ton of clients.

00:02:12.790 --> 00:02:14.900
Let me draw some of them.

00:02:14.900 --> 00:02:16.270
His eyes were strange.

00:02:16.270 --> 00:02:21.120
OK, so this broker has
a ton of clients,

00:02:21.120 --> 00:02:23.450
many of whom own IBM.

00:02:23.450 --> 00:02:26.590
And they keep their shares
of IBM with the broker.

00:02:26.590 --> 00:02:30.500
So the broker always has-- let's
say each of these is a

00:02:30.500 --> 00:02:31.750
share of IBM.

00:02:34.310 --> 00:02:36.280
And these guys are all
owners of IBM.

00:02:36.280 --> 00:02:38.960
And if ever, one day, this guy
wanted to sell his share of

00:02:38.960 --> 00:02:42.930
IBM, the broker would go out
there, sell it in the market,

00:02:42.930 --> 00:02:43.800
or maybe this guy
would sell it,

00:02:43.800 --> 00:02:45.030
depending on how it's done.

00:02:45.030 --> 00:02:47.360
And then this would be replaced
with the money, and

00:02:47.360 --> 00:02:50.010
this would then become
$100, right?

00:02:50.010 --> 00:02:52.890
And then if IBM issues a
dividend, if it says OK, all

00:02:52.890 --> 00:02:55.580
of our shareholders will get
$5.00 from a company.

00:02:55.580 --> 00:02:58.180
And then each of these
guys would get $5.00.

00:02:58.180 --> 00:03:00.650
So they're all just shareholders
in the company.

00:03:00.650 --> 00:03:03.980
And also, just so you know, as
opposed to being short the

00:03:03.980 --> 00:03:05.080
company, they're long.

00:03:05.080 --> 00:03:06.070
When you go long [? on a ?]

00:03:06.070 --> 00:03:07.970
security, you're buying
the security.

00:03:07.970 --> 00:03:11.320
When you go short, as you'll
see, you're borrowing the

00:03:11.320 --> 00:03:13.300
security and selling it.

00:03:13.300 --> 00:03:14.340
So let's see how that works.

00:03:14.340 --> 00:03:16.890
So once again I go to my broker
and I say broker, I

00:03:16.890 --> 00:03:19.040
want to borrow a share of IBM.

00:03:19.040 --> 00:03:21.160
And the broker says OK, well
I've got all these share

00:03:21.160 --> 00:03:23.000
sitting around here.

00:03:23.000 --> 00:03:25.010
And they're really not
doing anything.

00:03:25.010 --> 00:03:29.000
I'll lend you this share
right there.

00:03:29.000 --> 00:03:31.050
And stick it in my account.

00:03:31.050 --> 00:03:34.170
And I'm going to have to pay
rent to borrow that share.

00:03:34.170 --> 00:03:38.580
You can kind of view it as
either interest on borrowing

00:03:38.580 --> 00:03:39.690
the share, or you can
view it as rent.

00:03:39.690 --> 00:03:41.110
Rent is sometimes
more intuitive.

00:03:41.110 --> 00:03:45.050
Because whenever we talk about
when you rent someone's

00:03:45.050 --> 00:03:47.390
apartment, you're really kind of
borrowing their apartment.

00:03:47.390 --> 00:03:49.130
Then you can kind of view
the rent as the

00:03:49.130 --> 00:03:50.600
interest on the apartment.

00:03:50.600 --> 00:03:52.840
Or the other way to view it,
when you borrow money, the

00:03:52.840 --> 00:03:55.250
interest is kind of
rent on the money.

00:03:55.250 --> 00:03:56.690
Anyway, I think you
get the idea.

00:03:56.690 --> 00:03:57.660
But I'll borrow this.

00:03:57.660 --> 00:03:59.730
And you'll say hey wait, Sal,
that doesn't make sense.

00:03:59.730 --> 00:04:03.430
How can you just borrow
this guy's stock?

00:04:03.430 --> 00:04:05.710
I mean, this guy might want
to sell it the next day.

00:04:05.710 --> 00:04:06.980
And that's a good question.

00:04:06.980 --> 00:04:09.470
If this guy wants to-- let's say
I borrow his stock, it's

00:04:09.470 --> 00:04:11.220
not sitting there anymore.

00:04:11.220 --> 00:04:15.370
If this guy wants to sell his
stock the second after I

00:04:15.370 --> 00:04:17.760
borrow it, the broker is just
going to take-- he's just

00:04:17.760 --> 00:04:19.360
going to shuffle around the
stocks a little bit.

00:04:19.360 --> 00:04:22.019
I mean, you know stocks are--
they call it fungible, you can

00:04:22.019 --> 00:04:23.770
replace one stock certificate
with the other.

00:04:23.770 --> 00:04:24.860
They're no different.

00:04:24.860 --> 00:04:26.540
So then he'll say OK,
I'll just give this

00:04:26.540 --> 00:04:28.150
stock to this guy.

00:04:28.150 --> 00:04:29.310
Whoops.

00:04:29.310 --> 00:04:30.710
I just messed up with my pen.

00:04:30.710 --> 00:04:33.600
I'll just give this stock to the
guy who wants to sell it,

00:04:33.600 --> 00:04:34.610
and then you would
have essentially

00:04:34.610 --> 00:04:35.870
borrowed from this guy.

00:04:35.870 --> 00:04:38.940
And you could keep rearranging
the securities so that none of

00:04:38.940 --> 00:04:40.740
these guys ever know that
their stock is borrowed.

00:04:40.740 --> 00:04:42.550
Although you have to give
permission, especially if

00:04:42.550 --> 00:04:44.300
these guys own the
stock outright.

00:04:44.300 --> 00:04:46.870
They have to give permission to
let the stock be borrowed,

00:04:46.870 --> 00:04:49.910
and they benefit, because they
get some cut, hopefully, they

00:04:49.910 --> 00:04:52.960
get some cut of the rent, or the
interest that I'm paying.

00:04:52.960 --> 00:04:56.640
So it's beneficial to them.

00:04:56.640 --> 00:05:00.940
And the broker can never, well
it should never, really, lend

00:05:00.940 --> 00:05:02.300
out all of the shares here.

00:05:02.300 --> 00:05:04.290
Just so that it can keep
shuffling around.

00:05:04.290 --> 00:05:06.750
And we'll talk about what
happens if you actually try to

00:05:06.750 --> 00:05:08.360
do this with a share that
you've never borrowed.

00:05:08.360 --> 00:05:09.620
We'll do that in
a future thing.

00:05:09.620 --> 00:05:11.960
But the general idea is, you
borrow one of these shares out

00:05:11.960 --> 00:05:14.570
here, and then the broker can
kind of keep shuffling around

00:05:14.570 --> 00:05:16.780
these shares, if any of these
guys really want to get rid of

00:05:16.780 --> 00:05:17.460
their share.

00:05:17.460 --> 00:05:19.410
And let's say I've borrowed
this guy's.

00:05:19.410 --> 00:05:21.930
Let's say I originally borrowed
this guy's share.

00:05:21.930 --> 00:05:24.360
This guy wanted to
sell his share.

00:05:24.360 --> 00:05:27.480
So what the broker does is, he
takes this share, gives it to

00:05:27.480 --> 00:05:31.430
him, and now I have officially
borrowed this guy's share.

00:05:31.430 --> 00:05:32.170
Right?

00:05:32.170 --> 00:05:33.770
This is the share I'm borrowing
now because this

00:05:33.770 --> 00:05:35.490
one's been sold.

00:05:35.490 --> 00:05:39.070
Now, what happens if IBM
issues a dividend?

00:05:39.070 --> 00:05:43.190
It says every one of our
shareholders should get $5.00.

00:05:43.190 --> 00:05:47.240
Well this guy's expecting
a $5.00 check.

00:05:47.240 --> 00:05:49.130
Or he expects $5.00
to be deposited in

00:05:49.130 --> 00:05:50.600
his brokerage account.

00:05:50.600 --> 00:05:52.760
For that to happen, I
essentially have to write this

00:05:52.760 --> 00:05:55.160
guy the $5.00 check.

00:05:55.160 --> 00:05:58.770
So I have to make it look, or
at least the broker is going

00:05:58.770 --> 00:06:01.700
to force me to make it
look like this guy

00:06:01.700 --> 00:06:03.460
still owns the share.

00:06:03.460 --> 00:06:07.320
So anything that this guy gets
by being a shareholder of this

00:06:07.320 --> 00:06:10.240
stock, I have to provide
for him.

00:06:10.240 --> 00:06:11.740
Anyway, so I've borrowed
the stock.

00:06:11.740 --> 00:06:12.530
Now what do I do with it?

00:06:12.530 --> 00:06:15.805
My whole thesis here is that IBM
is going to $50, so how do

00:06:15.805 --> 00:06:17.420
I make money off of that?

00:06:17.420 --> 00:06:17.980
So I [? set ?]

00:06:17.980 --> 00:06:18.060
[? this ?]

00:06:18.060 --> 00:06:19.570
trading right now at $100.

00:06:19.570 --> 00:06:22.810
I've borrowed this share.

00:06:22.810 --> 00:06:23.610
So I sell it.

00:06:23.610 --> 00:06:30.590
So let's see, the first thing is
I borrow one share, as I've

00:06:30.590 --> 00:06:31.390
done there.

00:06:31.390 --> 00:06:34.790
And now I sell it
on the market.

00:06:34.790 --> 00:06:38.760
And we said it's selling
for $100.

00:06:38.760 --> 00:06:40.160
I sell it for $100.

00:06:40.160 --> 00:06:43.010
So I sell it to the market.

00:06:43.010 --> 00:06:44.580
Let's say the market is
out here to the left.

00:06:44.580 --> 00:06:48.980
So I sell to the market,
and I get $100.

00:06:48.980 --> 00:06:52.700
So let's see what's happening
right now.

00:06:52.700 --> 00:06:54.420
What's my current state
of affairs?

00:06:54.420 --> 00:06:57.900
Let's say assets-- and I'll talk
about margin requirements

00:06:57.900 --> 00:07:00.660
later-- liabilities.

00:07:00.660 --> 00:07:03.070
What do I have, what do I owe?

00:07:03.070 --> 00:07:05.650
So now I have $100.

00:07:05.650 --> 00:07:06.820
And what do I owe?

00:07:06.820 --> 00:07:12.920
I owe one share of IBM.

00:07:12.920 --> 00:07:15.850
Let's say my analysis ends
up being correct,

00:07:15.850 --> 00:07:17.100
often times it won't.

00:07:17.100 --> 00:07:19.210
And we'll talk a little bit more
about why short selling

00:07:19.210 --> 00:07:20.310
can be extremely risky.

00:07:20.310 --> 00:07:23.030
But let's say my analysis is
correct, and a couple of days

00:07:23.030 --> 00:07:26.270
later IBM starts
trading at $50.

00:07:26.270 --> 00:07:30.790
What I can do now is unwind my
trade, or cover my short.

00:07:30.790 --> 00:07:34.460
And the way I do this, is
I have $100, right?

00:07:34.460 --> 00:07:38.120
What I can do is, I can then
just go out there and buy my

00:07:38.120 --> 00:07:40.340
shares of IBM back, right?

00:07:40.340 --> 00:07:44.180
So I buy one share of IBM.

00:07:44.180 --> 00:07:45.060
And how much do I have to pay?

00:07:45.060 --> 00:07:46.330
Do I have to pay $100?

00:07:46.330 --> 00:07:49.265
Well no, the share of
IBM stock price

00:07:49.265 --> 00:07:51.190
has dropped to $50.

00:07:51.190 --> 00:07:52.850
So actually, let me do that.

00:07:52.850 --> 00:08:00.100
I'm not paying $100 out, I'm
going to pay $50 out.

00:08:00.100 --> 00:08:03.760
And I'm going to get back
a share of IBM.

00:08:03.760 --> 00:08:06.810
I'm going to get back a share of
IBM, and that's what I owe.

00:08:06.810 --> 00:08:10.510
So I'll give it back
to my brokerage.

00:08:10.510 --> 00:08:18.370
So I will go out, spend $50,
get back a share, and then

00:08:18.370 --> 00:08:20.420
give that share back to
the brokerage, give it

00:08:20.420 --> 00:08:21.560
back to that guy.

00:08:21.560 --> 00:08:23.760
These guys never knew that
anything happened.

00:08:23.760 --> 00:08:25.070
And in the process,
what happened?

00:08:25.070 --> 00:08:28.660
I sold the shares for
$100, and I bought

00:08:28.660 --> 00:08:31.030
them back for $50.

00:08:31.030 --> 00:08:33.470
So after unwinding this whole
thing-- I used $50 of this

00:08:33.470 --> 00:08:35.450
$100 to buy the shares back.

00:08:35.450 --> 00:08:36.450
So how much do I have left?

00:08:36.450 --> 00:08:39.390
I have $50 left.

00:08:39.390 --> 00:08:43.210
And I returned the shares of
IBM, so I owe no one anything.

00:08:43.210 --> 00:08:45.920
So I made $50 off
of this trade.

00:08:45.920 --> 00:08:49.160
So traditionally in the stock
market, on the long side you

00:08:49.160 --> 00:08:51.085
want to buy low, sell
high, right?

00:08:57.830 --> 00:09:00.450
When you're short selling,
you're doing the same thing,

00:09:00.450 --> 00:09:01.440
but you're doing
it in reverse.

00:09:01.440 --> 00:09:09.050
You want to sell high
and buy low.

00:09:09.050 --> 00:09:11.260
Anyway, hopefully that gives
you the general idea of the

00:09:11.260 --> 00:09:12.790
mechanics of short selling.

00:09:12.790 --> 00:09:15.600
In the next video I'll talk a
little bit more about what--

00:09:15.600 --> 00:09:18.780
this was a very good scenario,
where what I wanted to happen

00:09:18.780 --> 00:09:19.260
did happen.

00:09:19.260 --> 00:09:21.690
But in the next video we'll talk
about the risks of short

00:09:21.690 --> 00:09:24.935
selling, and maybe a little bit
of a discussion of whether

00:09:24.935 --> 00:09:28.240
it really is good or bad
for the markets, or for

00:09:28.240 --> 00:09:29.380
society as a whole.

00:09:29.380 --> 00:09:31.220
See you in the next video.


Poisson process 2

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.410
I think we now have all the
tools we need to move forward,

00:00:03.410 --> 00:00:05.590
so just to review a little bit
of the last video we said we

00:00:05.590 --> 00:00:08.680
are trying to model out the
probability distribution of

00:00:08.680 --> 00:00:10.780
how many cars might
pass in an hour.

00:00:10.780 --> 00:00:13.950
And the first thing we did is
we sat at that intersection and

00:00:13.950 --> 00:00:18.910
we found a pretty good expected
value of our random variable.

00:00:18.910 --> 00:00:21.720
And this random variable, just
to go back to the top, we

00:00:21.720 --> 00:00:24.030
defined the random variable as
the number of cars that pass

00:00:24.030 --> 00:00:28.790
in an hour at a certain
point on a certain road.

00:00:28.790 --> 00:00:31.410
We measure a bunch, we sat out
there a bunch of hours, and we

00:00:31.410 --> 00:00:34.540
got a pretty good estimate of
this and we say it's lambda.

00:00:34.540 --> 00:00:36.230
And we said, OK, we
wanted to model it as a

00:00:36.230 --> 00:00:37.440
binomial distribution.

00:00:37.440 --> 00:00:41.730
So if this is a binomial
distribution then this lambda

00:00:41.730 --> 00:00:45.840
would be equal to the number of
trials times the probability

00:00:45.840 --> 00:00:47.740
of success per trial.

00:00:47.740 --> 00:00:51.120
And so, if we could view a
trial as an interval of time.

00:00:51.120 --> 00:00:53.740
This is the total number
of successes in an hour.

00:00:57.320 --> 00:01:02.430
And so this would be success
in a smaller interval.

00:01:02.430 --> 00:01:04.820
And this would be the
probability of success in

00:01:04.820 --> 00:01:06.020
that smaller interval.

00:01:06.020 --> 00:01:07.540
And in the last video
we tried it out.

00:01:07.540 --> 00:01:10.460
We said, oh, well, what if we
make this interval a minute

00:01:10.460 --> 00:01:12.790
and this is the probability
of success per minute?

00:01:12.790 --> 00:01:15.280
We'd have maybe a reasonable
description of what

00:01:15.280 --> 00:01:15.850
we're describing.

00:01:15.850 --> 00:01:17.890
But what if more than one
car passes in a minute?

00:01:17.890 --> 00:01:19.840
And they said, oh, let's make
this per second and this

00:01:19.840 --> 00:01:21.680
is the probability of
success per second.

00:01:21.680 --> 00:01:23.130
But then we still have the
problem, more than one

00:01:23.130 --> 00:01:26.020
car could pass in a
second, very easily.

00:01:26.020 --> 00:01:29.020
So what we wanted to do is we
want to take the limit as this

00:01:29.020 --> 00:01:34.530
approaches infinity and then
see what kind of formula we

00:01:34.530 --> 00:01:38.610
get from the math gods.

00:01:38.610 --> 00:01:41.580
If we describes this as a
binomial distribution with the

00:01:41.580 --> 00:01:43.910
limit as it approaches
infinity, we could say that the

00:01:43.910 --> 00:01:48.940
probability that x is equal to
some number-- so the

00:01:48.940 --> 00:01:53.550
probability that our random
variable is equal to 3 cars in

00:01:53.550 --> 00:01:58.960
a particular hour, exactly 3
cars in an hour-- is equal to--

00:01:58.960 --> 00:02:00.460
oh, we want to take the limit
as it approaches

00:02:00.460 --> 00:02:01.430
infinity, right?

00:02:01.430 --> 00:02:09.090
The limit as n approaches
infinity of n choose k.

00:02:09.090 --> 00:02:12.050
We're going to have k moments
in time because n approaches

00:02:12.050 --> 00:02:16.250
infinity, these intervals
become super super duper small.

00:02:16.250 --> 00:02:20.170
So these become
moments in time.

00:02:20.170 --> 00:02:22.500
We're going to have close to an
infinite number moments and

00:02:22.500 --> 00:02:24.260
this is the number of
successful moments

00:02:24.260 --> 00:02:25.030
where cars pass.

00:02:25.030 --> 00:02:29.815
We have 3 moments where there
was a success, where a

00:02:29.815 --> 00:02:32.370
car passed, and we had
a total of 3 cars pass.

00:02:32.370 --> 00:02:35.960
Or 7 cars, 7 moments where it
was true that a car passed,

00:02:35.960 --> 00:02:38.760
and we would have total
7 cars pass in the hour.

00:02:38.760 --> 00:02:41.660
So just finishing up with our
binomial distribution, n

00:02:41.660 --> 00:02:44.890
moments, choose k successes
times the probability

00:02:44.890 --> 00:02:45.830
of success.

00:02:45.830 --> 00:02:47.220
What's the probability
of success?

00:02:52.000 --> 00:02:52.960
So this would be n.

00:02:52.960 --> 00:02:54.420
What's p equal to?

00:02:54.420 --> 00:02:57.980
p is equal to lambda divided by
n, right? n times p is lambda,

00:02:57.980 --> 00:03:02.250
so let me just write that down.
p is equal to lambda

00:03:02.250 --> 00:03:03.550
divided by n.

00:03:03.550 --> 00:03:06.530
I just rearranged this up here.

00:03:06.530 --> 00:03:09.380
So our probability of
success is lambda times n.

00:03:13.370 --> 00:03:14.650
And we're saying what's
the probability that

00:03:14.650 --> 00:03:15.650
we have k successes?

00:03:19.210 --> 00:03:20.990
And then, what's the
probability that

00:03:20.990 --> 00:03:22.380
we have a failure?

00:03:22.380 --> 00:03:25.180
Well, it's 1 minus the
probability of success.

00:03:25.180 --> 00:03:26.710
And how many failures
are we going to have?

00:03:26.710 --> 00:03:29.090
How many moments will
not have a car pass?

00:03:29.090 --> 00:03:31.420
Well, we have total event
moments and k of them were

00:03:31.420 --> 00:03:36.640
successes, so we'll have
n minus k failures.

00:03:36.640 --> 00:03:38.380
Let's see what we
can do with this.

00:03:38.380 --> 00:03:40.700
So this is equal to--
let me rewrite it all.

00:03:40.700 --> 00:03:42.790
And I'll change colors.

00:03:42.790 --> 00:03:47.840
The limit as n approaches
infinity-- let me write out

00:03:47.840 --> 00:03:49.140
this binomial coefficient.

00:03:49.140 --> 00:03:57.370
That's n factorial over n minus
k factorial times k factorial.

00:03:57.370 --> 00:03:58.530
Normally I write these the
other way around, but

00:03:58.530 --> 00:04:00.240
it's the same thing.

00:04:00.240 --> 00:04:05.750
Times lambda to the k.

00:04:05.750 --> 00:04:09.320
Using my exponent properties--
over n to the k.

00:04:09.320 --> 00:04:11.300
And then this expression
right here, I can actually

00:04:11.300 --> 00:04:12.460
separate out the exponents.

00:04:12.460 --> 00:04:16.790
This is the same thing as 1
minus lambda over n to the

00:04:16.790 --> 00:04:21.380
n times 1 minus lambda
over n to the minus k.

00:04:21.380 --> 00:04:23.130
You have the same base, you
could add the exponents and

00:04:23.130 --> 00:04:24.890
you would get this up here.

00:04:24.890 --> 00:04:27.030
Let me simplify a
little bit more.

00:04:27.030 --> 00:04:31.150
Let me swap spots
with these two.

00:04:31.150 --> 00:04:33.390
You can kind of view them both
as being in the denominator.

00:04:33.390 --> 00:04:36.630
So you can change the order of
division or multiplication

00:04:36.630 --> 00:04:37.490
depending how you view it.

00:04:37.490 --> 00:04:41.820
So this is equal to the
limit-- let me switch colors.

00:04:41.820 --> 00:04:44.050
The limit as n approaches
infinity-- I don't

00:04:44.050 --> 00:04:44.755
like that color.

00:04:48.180 --> 00:04:50.480
Actually, let me just rewrite
what we did in the last video.

00:04:50.480 --> 00:04:52.170
What is this thing right here?

00:04:52.170 --> 00:04:54.950
And we showed at the end of the
last video. n factorial divided

00:04:54.950 --> 00:04:57.130
by n minus k factorial.

00:04:57.130 --> 00:05:04.750
It was n times n minus 1
times n minus 2, all the

00:05:04.750 --> 00:05:08.560
way to n minus k plus 1.

00:05:08.560 --> 00:05:13.360
If this was 7 over 7 minus
2 factorial we would

00:05:13.360 --> 00:05:16.760
have 7 times 6.

00:05:16.760 --> 00:05:19.710
And 6 is one more
than 7 minus 2.

00:05:19.710 --> 00:05:20.660
So that's where we got that.

00:05:20.660 --> 00:05:22.600
We did that in the last video
if you're getting confused.

00:05:22.600 --> 00:05:23.830
And we also said that
there's going to be

00:05:23.830 --> 00:05:25.800
exactly k terms here.

00:05:25.800 --> 00:05:29.080
So if you counted these as 1,
2, 3-- all the way, there's

00:05:29.080 --> 00:05:32.240
going to be k terms here.

00:05:32.240 --> 00:05:32.930
We took care of that.

00:05:32.930 --> 00:05:34.620
We just rewrote that.

00:05:34.620 --> 00:05:36.830
And I said I would switch these
two things around, so that's

00:05:36.830 --> 00:05:46.240
divided by n to the k times--
I'm just switching

00:05:46.240 --> 00:05:52.015
these-- lambda to the k
over k factorial.

00:05:55.460 --> 00:05:57.540
And then, what do we have here?

00:05:57.540 --> 00:06:00.120
We'll I can just rewrite that.

00:06:00.120 --> 00:06:01.600
This is continuing
the same line.

00:06:01.600 --> 00:06:05.830
1 minus lambda over n to
the n times 1 minus lambda

00:06:05.830 --> 00:06:08.620
over n to the minus k.

00:06:08.620 --> 00:06:11.450
Now we can take the limit.

00:06:11.450 --> 00:06:13.420
So what happens when
we take the limit?

00:06:13.420 --> 00:06:15.870
If you take the limit, this
is another property so you

00:06:15.870 --> 00:06:19.410
don't get too overwhelmed--
another property of limits.

00:06:19.410 --> 00:06:23.100
If I take the limit as x
approaches anything, a

00:06:23.100 --> 00:06:26.320
of f of x times g of x.

00:06:26.320 --> 00:06:30.150
That's equal to the limit
as x approaches a of f of

00:06:30.150 --> 00:06:35.230
x, times the limit as x
approaches a of g of x.

00:06:35.230 --> 00:06:37.460
So we could take each of these
limits in the product and then

00:06:37.460 --> 00:06:40.040
multiply them and then
we'll be all set.

00:06:40.040 --> 00:06:41.070
So let's do that.

00:06:41.070 --> 00:06:43.220
And I want to leave
this stuff up here.

00:06:43.220 --> 00:06:44.560
So first of all,
what's this limit?

00:06:44.560 --> 00:06:46.320
Let me write this out.

00:06:46.320 --> 00:06:49.390
And let me pick a
good color-- yellow.

00:06:49.390 --> 00:06:56.790
So we have the limit as
n approaches infinity.

00:06:56.790 --> 00:06:59.930
So this thing up here, this n
times n minus 1 times n minus

00:06:59.930 --> 00:07:02.200
2-- all the way down to n
minus k plus 1, what's

00:07:02.200 --> 00:07:02.840
it going to look like?

00:07:02.840 --> 00:07:04.020
It's going to be a
polynomial right?

00:07:04.020 --> 00:07:06.380
We're multiplying a bunch of--
well really, we're multiplying

00:07:06.380 --> 00:07:09.540
a bunch of binomials and
we're doing it k times.

00:07:09.540 --> 00:07:13.210
So the largest degree term
is going to be n to the k.

00:07:13.210 --> 00:07:15.390
It's going to be n to the
k plus something times

00:07:15.390 --> 00:07:16.380
n to the k minus 1.

00:07:16.380 --> 00:07:21.280
It's going to be this
big polynomial-- kth

00:07:21.280 --> 00:07:22.010
degree polynomial.

00:07:22.010 --> 00:07:24.690
And that's really all we need
to know for this derivation.

00:07:24.690 --> 00:07:27.800
So it's going to be n to the
k plus blah, blah, blah,

00:07:27.800 --> 00:07:29.610
blah, blah, blah, blah--
a bunch of other stuff.

00:07:29.610 --> 00:07:34.190
This thing when you multiply it
out, over-- we have this n to

00:07:34.190 --> 00:07:39.310
the k, that's this part of it.

00:07:39.310 --> 00:07:44.960
Times the limit as--
well actually, we

00:07:44.960 --> 00:07:45.540
don't have to worry.

00:07:45.540 --> 00:07:46.390
This is a constant.

00:07:46.390 --> 00:07:47.900
So we could actually
bring this out front.

00:07:47.900 --> 00:07:49.340
So we don't even have
to write a limit.

00:07:49.340 --> 00:07:53.800
So times lambda to the
k k over k factorial.

00:07:53.800 --> 00:07:57.300
There's no n here, so this is
a constant with respect to n.

00:07:57.300 --> 00:08:04.560
Times the limit as n approaches
infinity of 1 minus lambda

00:08:04.560 --> 00:08:12.580
over n to the n times 1 minus
lambda over n to the minus k.

00:08:12.580 --> 00:08:16.220
All right, I know you
can barely read this.

00:08:16.220 --> 00:08:17.820
So first of all,
what's this limit?

00:08:17.820 --> 00:08:19.700
The limit as n approaches
infinity of some polynomial

00:08:19.700 --> 00:08:22.410
where it's n to the kth power
plus blah, blah, blah, blah.

00:08:22.410 --> 00:08:24.080
Where all of these other
terms have a lower degree.

00:08:24.080 --> 00:08:25.460
This is the highest
degree term.

00:08:28.050 --> 00:08:31.050
So you have n to the k in the
numerator and you have n to

00:08:31.050 --> 00:08:32.350
the k in the denominator.

00:08:32.350 --> 00:08:33.720
So the highest degrees
are the same.

00:08:33.720 --> 00:08:35.820
The coefficients are 1,
so this limit is 1.

00:08:35.820 --> 00:08:37.450
Another way you could do it,
you could divide the numerator

00:08:37.450 --> 00:08:43.250
and the denominator by n to the
k and you would get-- this

00:08:43.250 --> 00:08:47.670
would just be 1 plus 1 over n
plus 1 over-- everything else

00:08:47.670 --> 00:08:50.460
would have a 1 over n in it,
and this would just be a 1.

00:08:50.460 --> 00:08:52.640
And if you took the limit as
you approach infinity, then all

00:08:52.640 --> 00:08:53.910
of these other terms would be
zero and you'd be

00:08:53.910 --> 00:08:55.250
left with 1/1.

00:08:55.250 --> 00:08:57.140
But either way, you have the
same degree in the top and the

00:08:57.140 --> 00:09:01.340
bottom, and their coefficients
are the same, so the limit as n

00:09:01.340 --> 00:09:04.100
approaches infinity of this is
1, which is a nice

00:09:04.100 --> 00:09:05.320
simplification.

00:09:05.320 --> 00:09:12.670
So you end up with 1 times
lambda k over k factorial.

00:09:12.670 --> 00:09:14.660
Now what's the limit as n
approaches infinity of

00:09:14.660 --> 00:09:16.300
this thing right here?

00:09:16.300 --> 00:09:19.120
1 minus lambda over n to the n.

00:09:19.120 --> 00:09:21.350
Well, in the last video we
showed that it would be--

00:09:21.350 --> 00:09:22.270
I'll write it right here.

00:09:22.270 --> 00:09:30.850
That the limit as n approaches
infinity of 1 plus a over n to

00:09:30.850 --> 00:09:34.560
the n is equal to e to the a.

00:09:34.560 --> 00:09:36.320
That's exactly what we have
you here, but instead of an

00:09:36.320 --> 00:09:38.675
a we have a minus lambda.

00:09:41.590 --> 00:09:46.850
So this is going to be equal
to e to the minus lambda.

00:09:46.850 --> 00:09:48.770
We have a minus lambda
instead of an a.

00:09:48.770 --> 00:09:51.070
And then finally, what's the
limit as n approaches infinity?

00:09:51.070 --> 00:09:52.840
Let me write it a
little bit neater.

00:09:52.840 --> 00:09:53.880
I'm just rewriting this term.

00:09:53.880 --> 00:10:00.590
1 minus lambda over n
to the minus k power.

00:10:00.590 --> 00:10:02.980
What happens as n
approaches infinity?

00:10:02.980 --> 00:10:06.200
Well, this term,
lambda's a constant.

00:10:06.200 --> 00:10:08.050
As this approaches infinity,
this term's going

00:10:08.050 --> 00:10:08.970
to approach 0.

00:10:08.970 --> 00:10:10.430
So you have 1 to the minus k.

00:10:10.430 --> 00:10:14.240
1 to any power is 1, so
that term becomes 1.

00:10:14.240 --> 00:10:16.840
So we have another 1 there.

00:10:16.840 --> 00:10:17.930
So there you have it.

00:10:17.930 --> 00:10:20.510
We're done.

00:10:20.510 --> 00:10:23.480
The probability that our random
variable, the number of cars

00:10:23.480 --> 00:10:26.150
that passes in an hour, is
equal to a particular number.

00:10:26.150 --> 00:10:28.880
You know, it's equal to
7 cars pass in an hour.

00:10:28.880 --> 00:10:34.470
Is equal to the limit as n
approaches infinity of n choose

00:10:34.470 --> 00:10:41.660
k times-- well, we said it was
lambda over n to the k

00:10:41.660 --> 00:10:45.600
successes times 1 minus
lambda over n to the

00:10:45.600 --> 00:10:46.900
n minus k failures.

00:10:46.900 --> 00:10:51.070
And we just showed that this is
equal to lambda to the kth

00:10:51.070 --> 00:10:57.010
power over k factorial times
e to the minus lambda.

00:10:57.010 --> 00:10:59.680
And that's pretty neat because
when you just see it in kind of

00:10:59.680 --> 00:11:02.590
a vacuum-- if you have no
context for it, you wouldn't

00:11:02.590 --> 00:11:04.560
guess that this is in any
way related to the

00:11:04.560 --> 00:11:05.340
binomial theorem.

00:11:05.340 --> 00:11:07.000
I mean, it's got an e in there.

00:11:07.000 --> 00:11:09.890
It's got a factorial, but a lot
of things have factorials in

00:11:09.890 --> 00:11:12.170
life, so not clear that
that would make it

00:11:12.170 --> 00:11:12.750
a binomial theorem.

00:11:12.750 --> 00:11:15.445
But this is just the limit as
you take smaller and smaller

00:11:15.445 --> 00:11:17.730
and smaller intervals, and the
probability of success in each

00:11:17.730 --> 00:11:18.680
interval becomes smaller.

00:11:18.680 --> 00:11:20.620
But as you take the limit
you end up with e.

00:11:20.620 --> 00:11:23.410
And if you think about it it
makes sense because one of our

00:11:23.410 --> 00:11:25.660
derivations of e actually came
out of compound interest and we

00:11:25.660 --> 00:11:27.190
kind of did something
similar there.

00:11:27.190 --> 00:11:29.620
We took smaller and smaller
intervals of compounding and

00:11:29.620 --> 00:11:31.650
over each interval we
compounded by a much

00:11:31.650 --> 00:11:32.220
smaller number.

00:11:32.220 --> 00:11:34.060
And when you took the
limit you got e again.

00:11:34.060 --> 00:11:38.330
And that's actually where
that whole formula up here

00:11:38.330 --> 00:11:39.960
came from to begin with.

00:11:39.960 --> 00:11:42.940
But anyway, just so that you
know how to use this thing.

00:11:42.940 --> 00:11:46.080
So let's say that I were to go
out, I'm the traffic engineer,

00:11:46.080 --> 00:11:49.850
and I figure out that
on average, 9 cars

00:11:49.850 --> 00:11:50.810
passed per hour.

00:11:53.950 --> 00:11:58.570
And I want to know the
probability that-- so this

00:11:58.570 --> 00:11:59.920
is my expected value.

00:11:59.920 --> 00:12:02.910
In a given hour, on average,
9 cars are passing.

00:12:02.910 --> 00:12:06.100
So I want to know the
probability that 2 cars

00:12:06.100 --> 00:12:09.110
pass in a given hour,
exactly 2 cars pass.

00:12:09.110 --> 00:12:14.290
That's going to be equal to 9
cars per hour to the 2'th power

00:12:14.290 --> 00:12:16.850
or squared, to the 2'th power.

00:12:16.850 --> 00:12:23.030
Divided by 2 factorial times
e to the minus 9 power.

00:12:23.030 --> 00:12:31.650
So it's equal to 81 over 2
times e to the minus 9 power.

00:12:31.650 --> 00:12:35.010
And let's see, maybe I should
just get the graphing

00:12:35.010 --> 00:12:35.810
calculator out.

00:12:35.810 --> 00:12:38.170
Well, I'll let you do that
exercise to figure out what

00:12:38.170 --> 00:12:41.220
that is, but I'll see
you in the next video.


Poisson process 1

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:03.540
Let's say you're some type of
traffic engineer and what

00:00:03.540 --> 00:00:06.810
you're trying to figure out is,
how many cars pass by a certain

00:00:06.810 --> 00:00:08.320
point on the street at
any given point in time?

00:00:08.320 --> 00:00:10.210
And you want to figure out
the probabilities that a

00:00:10.210 --> 00:00:14.010
hundred cars pass or 5
cars pass in a given hour.

00:00:14.010 --> 00:00:15.810
So a good place to start is
just to define a random

00:00:15.810 --> 00:00:20.530
variable that essentially
represents what you care about.

00:00:20.530 --> 00:00:27.350
So let's say the number of cars
that pass in some amount of

00:00:27.350 --> 00:00:28.730
time, let's say, in an hour.

00:00:31.710 --> 00:00:34.510
And your goal is to figure out
the probability distribution of

00:00:34.510 --> 00:00:37.050
this random variable and then
once you know the probability

00:00:37.050 --> 00:00:39.450
distribution then you can
figure out what's the

00:00:39.450 --> 00:00:41.790
probability that 100 cars pass
in an hour or the probability

00:00:41.790 --> 00:00:45.890
that no cars pass in an hour
and you'd be unstoppable.

00:00:45.890 --> 00:00:48.290
And just a little aside, just
to move forward with this

00:00:48.290 --> 00:00:50.540
video, there's two assumptions
we need to make because

00:00:50.540 --> 00:00:52.235
we're going to study the
Poisson distribution.

00:00:52.235 --> 00:00:54.110
And in order to study it's
there's two assumptions

00:00:54.110 --> 00:00:54.630
we have to make.

00:00:54.630 --> 00:00:58.770
That Poisson hour at this point
on the street is no different

00:00:58.770 --> 00:00:59.650
than any other hour.

00:00:59.650 --> 00:01:01.340
And we know that that's
probably false.

00:01:01.340 --> 00:01:03.750
During rush hour in a real
situation you probably

00:01:03.750 --> 00:01:06.640
would have more cars than
at another rush hour.

00:01:06.640 --> 00:01:08.640
And you know, if you wanted to
be more realistic maybe we do

00:01:08.640 --> 00:01:12.370
it in the day because in a day
any period of time--

00:01:12.370 --> 00:01:12.750
actually, no.

00:01:12.750 --> 00:01:14.120
I shouldn't do a day.

00:01:14.120 --> 00:01:17.750
We have to assume that every
hour is completely just like

00:01:17.750 --> 00:01:19.650
any other hour and actually,
even within the hour there's

00:01:19.650 --> 00:01:22.990
really no differentiation from
one second to the other in

00:01:22.990 --> 00:01:25.820
terms of the probabilities
that a car arrives.

00:01:25.820 --> 00:01:27.950
That's a little bit of a
simplifying assumption that

00:01:27.950 --> 00:01:29.950
might not truly apply to
traffic, but I think we

00:01:29.950 --> 00:01:32.270
can make that assumption.

00:01:32.270 --> 00:01:34.160
And then the other assumption
we need to make is that if a

00:01:34.160 --> 00:01:36.690
bunch of cars pass in one hour
that doesn't mean that fewer

00:01:36.690 --> 00:01:37.820
cars will pass in the next.

00:01:37.820 --> 00:01:40.630
That in no way does the number
of cars that pass in one period

00:01:40.630 --> 00:01:44.860
affect or correlate or somehow
influence the number of cars

00:01:44.860 --> 00:01:45.380
that pass in the next.

00:01:45.380 --> 00:01:47.370
That they're really
independent.

00:01:47.370 --> 00:01:50.670
Given that, we can then at
least try using the skills

00:01:50.670 --> 00:01:53.480
we have to model out some
type of a distribution.

00:01:53.480 --> 00:01:55.770
The first thing you do and I'd
recommend doing this for any

00:01:55.770 --> 00:01:59.090
distribution is maybe we
can estimate the mean.

00:01:59.090 --> 00:02:03.040
Let's sit out on that curb and
measure what this variable is

00:02:03.040 --> 00:02:05.170
over a bunch of hours and then
average it up, and that's going

00:02:05.170 --> 00:02:08.890
to be a pretty good estimator
for the actual mean

00:02:08.890 --> 00:02:09.880
of our population.

00:02:09.880 --> 00:02:12.270
Or, since it's a random
variable, the expected value

00:02:12.270 --> 00:02:13.010
of this random variable.

00:02:13.010 --> 00:02:16.660
Let's say you do that and you
get your best estimate of the

00:02:16.660 --> 00:02:22.270
expected value of this random
variable is-- I'll use

00:02:22.270 --> 00:02:24.850
the letter lambda.

00:02:24.850 --> 00:02:27.380
You know, this could
be 9 cars per hour.

00:02:27.380 --> 00:02:30.190
You sat out there-- it could
be 9.3 cars per hour.

00:02:30.190 --> 00:02:32.670
You sat out there over hundreds
of hours and you just counted

00:02:32.670 --> 00:02:34.590
the number of cars each hour
and you averaged them all up.

00:02:34.590 --> 00:02:37.250
You said, on average, there are
9.3 cars per hour and you feel

00:02:37.250 --> 00:02:38.680
that's a pretty good estimate.

00:02:38.680 --> 00:02:40.080
So that's what you have there.

00:02:40.080 --> 00:02:42.000
And let's see what we could do.

00:02:42.000 --> 00:02:45.560
We know the binomial
distribution.

00:02:45.560 --> 00:02:50.650
The binomial distribution tells
us that the expected value of a

00:02:50.650 --> 00:02:55.220
random variable is equal to the
number of trials that that

00:02:55.220 --> 00:02:57.460
random variable's kind
of composed of, right?

00:02:57.460 --> 00:02:59.490
Before, in the previous videos
we were counting the number

00:02:59.490 --> 00:03:00.500
of heads in a coin toss.

00:03:00.500 --> 00:03:03.070
So this would be the number
of coin tosses, times the

00:03:03.070 --> 00:03:07.290
probability of success
over each toss.

00:03:07.290 --> 00:03:09.000
This is what we did with
the binomial distribution.

00:03:09.000 --> 00:03:11.670
So maybe we can model
our traffic situation

00:03:11.670 --> 00:03:12.780
something similar.

00:03:12.780 --> 00:03:15.400
This is the number of cars
that pass in an hour.

00:03:15.400 --> 00:03:22.800
So maybe we could say lambda
cars per hour is equal

00:03:22.800 --> 00:03:24.330
to-- I don't know.

00:03:26.850 --> 00:03:29.880
Let's make each experiment or
each toss of the coin equal to

00:03:29.880 --> 00:03:31.780
whether a car passes
in a given minute.

00:03:31.780 --> 00:03:37.980
So there are 60 minutes
per hour, so there

00:03:37.980 --> 00:03:40.870
would be 60 trials.

00:03:40.870 --> 00:03:43.190
And then, the probability that
we have success in each of

00:03:43.190 --> 00:03:46.990
those trials, if we modeled
this as a binomial distribution

00:03:46.990 --> 00:03:54.450
would be lambda over
60 cars per minute.

00:03:54.450 --> 00:03:55.660
And this would be
a probability.

00:03:55.660 --> 00:03:58.640
This would be n, and this would
be the probability, if we said

00:03:58.640 --> 00:04:00.270
that this is a binomial
distribution.

00:04:00.270 --> 00:04:04.030
And this probably wouldn't be
that bad of an approximation.

00:04:04.030 --> 00:04:06.130
If you actually then said,
oh, this is a binomial

00:04:06.130 --> 00:04:10.380
distribution, so the
probability that our random

00:04:10.380 --> 00:04:12.940
variable equals some
given value, k.

00:04:12.940 --> 00:04:16.170
You know, the probability that
3 cars, exactly 3 cars pass in

00:04:16.170 --> 00:04:19.750
an given hour, we would
then be equal to n.

00:04:19.750 --> 00:04:21.890
So n would be 60.

00:04:21.890 --> 00:04:26.010
Choose k, and you know,
I have 3 cars times the

00:04:26.010 --> 00:04:27.190
probability of success.

00:04:27.190 --> 00:04:29.570
So the probability that a
car passes in any minute.

00:04:29.570 --> 00:04:34.770
So it'd be lambda over
60 to the number of

00:04:34.770 --> 00:04:35.980
successes we need.

00:04:35.980 --> 00:04:41.660
So to the kth power, times the
probability of no success or

00:04:41.660 --> 00:04:46.560
that no cars pass,
to the n minus k.

00:04:46.560 --> 00:04:50.230
If we have k successes we have
to have 60 minus k failures.

00:04:50.230 --> 00:04:52.950
There are 60 minus k minutes
where no car passed.

00:04:52.950 --> 00:04:55.270
This actually wouldn't be that
bad of an approximation where

00:04:55.270 --> 00:04:57.250
you have 60 intervals and you
say this is a binomial

00:04:57.250 --> 00:04:58.560
distribution.

00:04:58.560 --> 00:05:00.310
And you'd probably get
reasonable results.

00:05:00.310 --> 00:05:02.600
But there's a core issue here.

00:05:02.600 --> 00:05:06.580
In this model where we model it
as a binomial distribution,

00:05:06.580 --> 00:05:09.980
what happens if more than
one car passes in an hour?

00:05:09.980 --> 00:05:11.630
Or more than one car
passes in a minute?

00:05:11.630 --> 00:05:14.270
The way we have it right now
we call it a success if one

00:05:14.270 --> 00:05:15.320
car passes in a minute.

00:05:15.320 --> 00:05:18.790
And if you're kind of counting
it counts as one success, even

00:05:18.790 --> 00:05:21.190
if 5 cars pass in that minute.

00:05:21.190 --> 00:05:23.390
So you say, oh, OK Sal, I
know the solution there.

00:05:23.390 --> 00:05:26.040
I just have to get
more granular.

00:05:26.040 --> 00:05:28.870
Instead of dividing it
into minutes why don't I

00:05:28.870 --> 00:05:31.050
divide it into seconds?

00:05:31.050 --> 00:05:36.210
So the probability that I have
k successes-- instead of 60

00:05:36.210 --> 00:05:39.820
intervals I'll do
3,600 intervals.

00:05:39.820 --> 00:05:43.170
So the probability of k
successful seconds, so a second

00:05:43.170 --> 00:05:48.610
where a car is passing at that
moment out of 3,600 seconds.

00:05:48.610 --> 00:05:52.190
So that's 3,600 choose k, times
the probability that a car

00:05:52.190 --> 00:05:55.210
passes in any given second.

00:05:55.210 --> 00:05:57.930
That's the expected number of
cars in an hour divided by

00:05:57.930 --> 00:06:00.430
number seconds in an hour.

00:06:00.430 --> 00:06:01.403
We're going to
have k successes.

00:06:03.990 --> 00:06:06.270
And these are the failures,
the probability of a failure

00:06:06.270 --> 00:06:12.050
and you're going to have
3,600 minus k failures.

00:06:12.050 --> 00:06:13.910
And this would be even a
better approximation.

00:06:13.910 --> 00:06:16.770
This actually would not be so
bad, but still, you have this

00:06:16.770 --> 00:06:19.100
situation where 2 cars
can come within a half a

00:06:19.100 --> 00:06:19.980
second of each other.

00:06:19.980 --> 00:06:21.910
And you say, oh, OK Sal,
I see the pattern here.

00:06:21.910 --> 00:06:23.650
We just have to get more
and more granular.

00:06:23.650 --> 00:06:26.170
We have to just make
this number larger and

00:06:26.170 --> 00:06:27.400
larger and larger.

00:06:27.400 --> 00:06:28.950
And your intuition is correct.

00:06:28.950 --> 00:06:31.340
And if you do that you'll
end up getting the

00:06:31.340 --> 00:06:33.860
Poisson distribution.

00:06:33.860 --> 00:06:35.620
And this is really interesting
because a lot of times people

00:06:35.620 --> 00:06:38.600
give you the formula for the
Poisson distribution and you

00:06:38.600 --> 00:06:40.420
can kind of just plug in
the numbers and use it.

00:06:40.420 --> 00:06:43.250
But it's neat to know that it
really is just the binomial

00:06:43.250 --> 00:06:45.790
distribution and the binomial
distribution really did come

00:06:45.790 --> 00:06:48.590
from kind of the common
sense of flipping coins.

00:06:48.590 --> 00:06:50.500
That's where everything
is coming from.

00:06:50.500 --> 00:06:53.710
But before we kind of prove
that if we take the limit

00:06:53.710 --> 00:06:55.670
as-- let me change colors.

00:06:55.670 --> 00:06:58.470
Before we proved that as we
take the limit as this number

00:06:58.470 --> 00:07:01.270
right here, the number of
intervals approaches infinity

00:07:01.270 --> 00:07:04.070
that this becomes the
Poisson distribution.

00:07:04.070 --> 00:07:07.290
I'm going to make sure we have
a couple of mathematical

00:07:07.290 --> 00:07:09.150
tools in our belt.

00:07:09.150 --> 00:07:12.760
So the first is something that
you're probably reasonably

00:07:12.760 --> 00:07:15.860
familiar with by now, but I
just want to make sure that the

00:07:15.860 --> 00:07:25.680
limit as x approaches infinity
of 1 plus a/x to the x power is

00:07:25.680 --> 00:07:31.020
equal to e to the
ax-- no sorry.

00:07:31.020 --> 00:07:38.020
Is equal to e to the a and now
just to prove this to you,

00:07:38.020 --> 00:07:39.260
let's make a little
substitution here.

00:07:39.260 --> 00:07:43.640
Let's say that n is equal
to-- let me say 1 over

00:07:43.640 --> 00:07:47.880
n is equal to a over x.

00:07:47.880 --> 00:07:52.890
And then what would be
x would equal to na.

00:07:52.890 --> 00:07:55.290
x times 1 is equal
to n times a.

00:07:55.290 --> 00:08:00.050
And so the limit as x
approaches infinity,

00:08:00.050 --> 00:08:02.045
what does a approach?

00:08:02.045 --> 00:08:02.885
a is-- sorry.

00:08:02.885 --> 00:08:04.920
As x approaches infinity
what does n approach?

00:08:04.920 --> 00:08:07.350
Well n is x divided by a.

00:08:07.350 --> 00:08:08.710
So n would also
approach infinity.

00:08:08.710 --> 00:08:10.810
So this thing would be the same
thing as just making our

00:08:10.810 --> 00:08:16.460
substitution the limit as n
approaches infinity of 1

00:08:16.460 --> 00:08:21.390
plus-- a/x, I made the
substitution as 1/n.

00:08:21.390 --> 00:08:26.720
And x is, by this
substitution, n times a.

00:08:26.720 --> 00:08:30.500
And this is going to be the
same thing as the limit as n

00:08:30.500 --> 00:08:36.090
approaches infinity of 1 plus
1/n to the n, all

00:08:36.090 --> 00:08:39.390
of that to the a.

00:08:39.390 --> 00:08:41.760
And since there's no n out here
we could just take the limit

00:08:41.760 --> 00:08:43.450
of this and then take
that to the a power.

00:08:43.450 --> 00:08:47.690
So that's going to be equal to
the limit as n approaches

00:08:47.690 --> 00:08:52.600
infinity of 1 plus 1/n to the
nth power, all of

00:08:52.600 --> 00:08:53.780
that to the a.

00:08:53.780 --> 00:08:58.040
And this is our definition, or
one of the ways to get to e if

00:08:58.040 --> 00:09:00.820
you'd watch the videos on
compound interest and all that.

00:09:00.820 --> 00:09:01.880
This is how we got to e.

00:09:01.880 --> 00:09:03.460
And if you tried it out on your
calculator, just try larger

00:09:03.460 --> 00:09:07.260
and larger n's here
and you'll get e.

00:09:07.260 --> 00:09:12.010
This inner part is equal to e,
and we raised it to the a

00:09:12.010 --> 00:09:14.060
power, so it's equal
to e to the a.

00:09:14.060 --> 00:09:16.240
So hopefully you pretty
satisfied that this limit

00:09:16.240 --> 00:09:17.860
is equal to e to the a.

00:09:17.860 --> 00:09:19.860
And then one other tool kit I
want in our belt, and I'll

00:09:19.860 --> 00:09:22.340
probably actually do the
proof in the next video.

00:09:22.340 --> 00:09:32.950
The other tool kit is to
recognize that x factorial over

00:09:32.950 --> 00:09:42.860
x minus k factorial is equal to
x times x minus 1 times x

00:09:42.860 --> 00:09:50.030
minus 2, all the way down
to times x minus k plus 1.

00:09:50.030 --> 00:09:51.880
And we've done this a lot of
times, but this is the most

00:09:51.880 --> 00:09:53.060
abstract we've ever written it.

00:09:53.060 --> 00:09:55.580
I can give you a couple of--
and just so you know, they'll

00:09:55.580 --> 00:09:57.330
be exactly k terms here.

00:09:57.330 --> 00:10:01.700
1, 2, 3-- So first term, second
term, third term, all the

00:10:01.700 --> 00:10:04.310
way, and this the kth term.

00:10:04.310 --> 00:10:07.210
And this is important to
our derivation of the

00:10:07.210 --> 00:10:09.160
Poisson distribution.

00:10:09.160 --> 00:10:13.870
But just to make this in real
numbers, if I had 7 factorial

00:10:13.870 --> 00:10:20.110
over 7 minus 2 factorial,
that's equal to 7 times 6

00:10:20.110 --> 00:10:24.070
times 5 times 4 times
3 times 3 times 1.

00:10:24.070 --> 00:10:27.360
Over 2 times-- no sorry.

00:10:27.360 --> 00:10:28.940
7 minus 2, this is 5.

00:10:28.940 --> 00:10:33.500
So it's over 5 times 4
times 3 times 2 times 1.

00:10:33.500 --> 00:10:37.190
These cancel out and you
just have 7 times 6.

00:10:37.190 --> 00:10:40.990
And so it's 7 and then
the last term is 7 minus

00:10:40.990 --> 00:10:43.045
2 plus 1, which is 6.

00:10:47.560 --> 00:10:51.290
In this example, k was 2 and
you had exactly 2 terms.

00:10:51.290 --> 00:10:53.230
So once we know those two
things we're now ready

00:10:53.230 --> 00:10:55.710
to derive the Poisson
distribution and I'll do

00:10:55.710 --> 00:10:58.415
that in the next video.

00:10:58.415 --> 00:10:59.980
See you soon.


Expected value of binomial distribution

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:02.430
In the last video we learned a
little bit about what the

00:00:02.430 --> 00:00:06.030
expected value of random
variable is, and we saw that it

00:00:06.030 --> 00:00:09.660
was really just the population
mean-- the same thing.

00:00:09.660 --> 00:00:12.600
But with a random variable,
since the population is

00:00:12.600 --> 00:00:14.880
infinite, you can't take
up all of the terms and

00:00:14.880 --> 00:00:16.140
then average them out.

00:00:16.140 --> 00:00:18.620
What you have to do is say OK,
each of these terms occur with

00:00:18.620 --> 00:00:21.280
some frequency or with some
probability, and you kind of

00:00:21.280 --> 00:00:23.100
just take a probability
weighted sum.

00:00:23.100 --> 00:00:25.180
Which we saw in the last video
was the exact same thing as

00:00:25.180 --> 00:00:27.420
adding everything together and
dividing by the number of

00:00:27.420 --> 00:00:30.460
numbers, except that that
method worked with an infinite

00:00:30.460 --> 00:00:32.850
number of an infinite
population what the

00:00:32.850 --> 00:00:33.510
random variable is.

00:00:33.510 --> 00:00:37.230
Because you can just keep on
performing the experiment that

00:00:37.230 --> 00:00:38.980
generates the random variable.

00:00:38.980 --> 00:00:41.400
And then, we actually
calculated the expected value

00:00:41.400 --> 00:00:46.580
for the particular binomial
distributions that we studied,

00:00:46.580 --> 00:00:50.040
especially the one with the
flipping of the coin.

00:00:50.040 --> 00:00:53.840
In this video we'll find a
general formula for the mean,

00:00:53.840 --> 00:00:57.200
or actually, for the expected
value of a binomial

00:00:57.200 --> 00:00:57.830
distribution.

00:00:57.830 --> 00:01:05.750
So if we say that the random
variable, x, is equal to the

00:01:05.750 --> 00:01:09.795
number of-- we could
call it successes.

00:01:12.550 --> 00:01:30.430
The number of successes with
probability p after n trials.

00:01:30.430 --> 00:01:31.770
So I'm being a little
bit generally here.

00:01:31.770 --> 00:01:37.060
I mean I could have said the
number of successful heads,

00:01:37.060 --> 00:01:39.940
which have a probability
of 0.5 after 10 flips.

00:01:39.940 --> 00:01:41.420
That's the same thing is this,
I'm just being a little

00:01:41.420 --> 00:01:42.990
bit more general here.

00:01:42.990 --> 00:01:44.300
And now, we'll actually
figure out what the

00:01:44.300 --> 00:01:47.500
expected value of this is.

00:01:47.500 --> 00:01:50.850
And we saw if you actually
figured out the probability

00:01:50.850 --> 00:01:52.970
distribution for this random
variable you get that nice

00:01:52.970 --> 00:01:55.070
binomial distribution that
looks a little bit

00:01:55.070 --> 00:01:55.840
like a bell curve.

00:01:55.840 --> 00:01:58.360
And we'll study more
about bell curves later.

00:01:58.360 --> 00:02:00.580
And before I actually
show it to you I'll

00:02:00.580 --> 00:02:01.980
give you the answer.

00:02:01.980 --> 00:02:03.470
Because the answer, to
some degree, is actually

00:02:03.470 --> 00:02:05.850
pretty intuitive.

00:02:05.850 --> 00:02:12.730
The expected value of this
random variable is n times

00:02:12.730 --> 00:02:15.810
p, or sometimes people
will write p times n.

00:02:15.810 --> 00:02:17.920
Let me make that a little
bit more tangible for you.

00:02:17.920 --> 00:02:21.600
So if I said that x is-- let me
do it in a different color.

00:02:21.600 --> 00:02:31.510
Let's see, x is equal to the
number of baskets I make.

00:02:31.510 --> 00:02:34.430
Where I'm talking about
basketball, not basket weaving.

00:02:34.430 --> 00:02:43.350
Number of baskets I make after
10 shots where I have the

00:02:43.350 --> 00:02:48.270
probability of me making any 1
shot is-- I don't know-- 40%.

00:02:48.270 --> 00:02:52.010
We know that the expected
number of baskets I

00:02:52.010 --> 00:02:53.550
make after 10 shots.

00:02:53.550 --> 00:02:57.120
So we know that the expected
number of baskets I make after

00:02:57.120 --> 00:03:00.770
10 shots, where each of my
shots is 40%-- all I have to do

00:03:00.770 --> 00:03:03.410
is multiply the probability
times the number of

00:03:03.410 --> 00:03:04.720
baskets I'm taking.

00:03:04.720 --> 00:03:09.390
So I multiply probability times
the number of baskets or the

00:03:09.390 --> 00:03:12.740
number of shots I'm taking,
which should be equal to 4.

00:03:12.740 --> 00:03:14.940
So I know I said-- and you
really shouldn't necessarily

00:03:14.940 --> 00:03:18.310
strictly view expected value as
the number of shots you should

00:03:18.310 --> 00:03:20.630
expect to make because
sometimes probability

00:03:20.630 --> 00:03:22.770
distributions can
be kind of weird.

00:03:22.770 --> 00:03:24.490
But in the binomial
distribution you can kind

00:03:24.490 --> 00:03:25.290
of view it that way.

00:03:25.290 --> 00:03:28.200
That this is the number of
shots you would expect to make.

00:03:28.200 --> 00:03:31.090
Or you can kind of view it
as the most likely outcome.

00:03:31.090 --> 00:03:35.340
That if you have a 40% shot
percentage, and you takes 10

00:03:35.340 --> 00:03:38.320
shots, the most likely outcome
is that you'll make 4 shots.

00:03:38.320 --> 00:03:41.080
You still might make 6 shots or
3 shots, but this is going to

00:03:41.080 --> 00:03:42.650
be the most likely outcome.

00:03:42.650 --> 00:03:44.790
and in my head, the way I think
about, the way it makes

00:03:44.790 --> 00:03:49.260
intuitive sense is that every
time you shoot you have a 40%

00:03:49.260 --> 00:03:51.280
chance of making the shot.

00:03:51.280 --> 00:03:55.120
So you could say that you
always make 40% of a shot.

00:03:55.120 --> 00:03:58.390
And if you take 10 shots you're
going to make 4 whole shots.

00:03:58.390 --> 00:04:00.420
So that's one way to think
about it and why this might

00:04:00.420 --> 00:04:02.190
make a little intuitive sense.

00:04:02.190 --> 00:04:05.170
But now, let's prove it to
ourselves that this is really

00:04:05.170 --> 00:04:10.110
true for any a random variable
that's described by a

00:04:10.110 --> 00:04:12.170
binomial distribution.

00:04:12.170 --> 00:04:15.270
So in a binomial distribution
what is the probability-- so if

00:04:15.270 --> 00:04:20.110
I say, what is the probability
that X is equal to k?

00:04:20.110 --> 00:04:21.770
And I know it just gets a
little complicated sometimes.

00:04:21.770 --> 00:04:24.010
But I'm just saying, what's
the probability say in

00:04:24.010 --> 00:04:25.440
this basketball analogy.

00:04:25.440 --> 00:04:29.600
Would be you know, what's the
probability that I make--

00:04:29.600 --> 00:04:31.850
k could be 3 shots or
something like that.

00:04:31.850 --> 00:04:32.880
So that's what we're
talking about.

00:04:32.880 --> 00:04:37.350
And that we learned was, if
we're taking n shots we're

00:04:37.350 --> 00:04:39.720
going to choose k of them.

00:04:39.720 --> 00:04:42.740
And we did that several times
in the last couple of videos.

00:04:42.740 --> 00:04:45.940
And then we multiply that times
the probability of any one of

00:04:45.940 --> 00:04:47.610
those particular occurrences.

00:04:47.610 --> 00:04:52.100
So if I'm making k shots, it'll
be the probability of me making

00:04:52.100 --> 00:04:56.610
any one shot, Which is
p to the kth power.

00:04:56.610 --> 00:04:57.950
p times itself k times.

00:04:57.950 --> 00:05:00.360
That's the probability
of me making k shots.

00:05:00.360 --> 00:05:02.800
And then the rest of the
shots I have to miss.

00:05:02.800 --> 00:05:06.360
So the probability of
a miss is 1 minus p.

00:05:06.360 --> 00:05:08.180
And then how many shots?

00:05:08.180 --> 00:05:11.450
If I've made k shots, the rest
of the shots I have to miss.

00:05:11.450 --> 00:05:14.780
So I'm going to miss
n minus k shots.

00:05:14.780 --> 00:05:18.240
So in any binomial distribution
this is a probability

00:05:18.240 --> 00:05:22.160
that you get k successes.

00:05:22.160 --> 00:05:24.380
Now we know that the expected
value, the way you calculate an

00:05:24.380 --> 00:05:27.020
expected value of a random
variable is you just take the

00:05:27.020 --> 00:05:29.710
probability weighted sum.

00:05:29.710 --> 00:05:33.160
I don't want to confuse you
too much and if you main take

00:05:33.160 --> 00:05:35.960
away from this video is just
this, that's good enough.

00:05:35.960 --> 00:05:36.990
You should feel good.

00:05:36.990 --> 00:05:38.890
Now it'll get a little
technical, but it'll hopefully

00:05:38.890 --> 00:05:40.630
make you a little bit more
comfortable with sigma and

00:05:40.630 --> 00:05:41.680
sum notation as well.

00:05:41.680 --> 00:05:43.570
It'll make you a little
bit more comfortable with

00:05:43.570 --> 00:05:46.250
binomial coefficients
and things like that.

00:05:46.250 --> 00:05:48.140
But just going back, the
expected value is a a

00:05:48.140 --> 00:05:50.130
probability weighted
sum of each of these.

00:05:50.130 --> 00:05:51.700
So what you want to do is you
want to take the probability

00:05:51.700 --> 00:05:56.300
that X is equal to k, times
k, and then add that up for

00:05:56.300 --> 00:05:57.580
each of the possible k's.

00:05:57.580 --> 00:05:58.860
So how would I write that?

00:05:58.860 --> 00:06:04.600
So the expected value of X, the
expected value of our random

00:06:04.600 --> 00:06:07.560
variable that's being described
as binomial distribution--

00:06:07.560 --> 00:06:08.340
it's equal to the sum.

00:06:10.910 --> 00:06:13.750
And we're going to sum all of
the values that k can take on.

00:06:13.750 --> 00:06:18.480
So k can start at 0-- in the
basketball version, I make no

00:06:18.480 --> 00:06:23.160
shots-- all the way to n,
which means I make n shots.

00:06:23.160 --> 00:06:27.350
And for each of them you want
to multiply k, so the outcome,

00:06:27.350 --> 00:06:29.390
so I made k shots, times the
probability that

00:06:29.390 --> 00:06:30.030
I make k shots.

00:06:30.030 --> 00:06:32.700
Well, what was the probability
that I make k shots?

00:06:32.700 --> 00:06:34.460
That was this right here.

00:06:34.460 --> 00:06:41.730
So it'd be k times n choose
k times p to the k times 1

00:06:41.730 --> 00:06:44.730
minus p to the n minus k.

00:06:44.730 --> 00:06:48.370
And now we're just going
to do some algebra, some

00:06:48.370 --> 00:06:50.750
sigma algebra I guess
you could call it.

00:06:50.750 --> 00:06:53.380
So the first simplification
we can make is we're summing

00:06:53.380 --> 00:06:54.710
from k equals 0 to n.

00:06:54.710 --> 00:06:57.240
So the first term here is going
to have a k equals 0 here.

00:06:57.240 --> 00:07:00.230
This is going to be 0
in the first term.

00:07:00.230 --> 00:07:03.350
So this first term is 0, then
this whole thing is going to be

00:07:03.350 --> 00:07:07.170
0, and the k equals 0 term
won't contribute to the sum

00:07:07.170 --> 00:07:09.910
because this whole
thing will be 0.

00:07:09.910 --> 00:07:12.800
Let me that write because I
think it's-- so this sum could

00:07:12.800 --> 00:07:20.450
be written as 0 times n choose
0 times p to the 0 times 1

00:07:20.450 --> 00:07:23.580
minus p to the n minus 0.

00:07:23.580 --> 00:07:28.600
Plus 1 times n choose 1 times
p to the 1 times 1 minus

00:07:28.600 --> 00:07:30.710
p to the n minus 1.

00:07:30.710 --> 00:07:33.150
And then you keep adding,
all the way until you

00:07:33.150 --> 00:07:34.810
get to k is equal to n.

00:07:34.810 --> 00:07:39.430
So it'd be n times n choose
n times p to the n, times

00:07:39.430 --> 00:07:42.250
1 minus p, n minus n.

00:07:42.250 --> 00:07:44.650
This is just another way of
writing this sum up here.

00:07:44.650 --> 00:07:47.690
And what I just said is the
first term, which is this term,

00:07:47.690 --> 00:07:49.920
is going to be equal to 0
because k is equal to 0.

00:07:49.920 --> 00:07:52.090
0 times any thing is 0.

00:07:52.090 --> 00:07:55.540
So we can ignore that term and
we can rewrite this sum as

00:07:55.540 --> 00:07:58.230
essentially this
sum right here.

00:08:02.020 --> 00:08:03.660
And so if we were to do that
we're essentially just

00:08:03.660 --> 00:08:04.820
rewrite this thing up here.

00:08:04.820 --> 00:08:08.730
So the expected value
of our random variable

00:08:08.730 --> 00:08:10.890
is equal to the sum.

00:08:10.890 --> 00:08:12.540
And we don't have to go
from k equal 0, we could

00:08:12.540 --> 00:08:14.450
start at k equal 1 now.

00:08:14.450 --> 00:08:20.130
From k equals 1 to n of the
same thing. k times n choose

00:08:20.130 --> 00:08:28.230
k times p to the k, times
1 minus p, n minus k.

00:08:28.230 --> 00:08:30.490
Let's see what we
can do from here.

00:08:30.490 --> 00:08:32.600
All we did so far is we got rid
of that first term because

00:08:32.600 --> 00:08:35.410
that's kind of the trick we'll
use to simplify this eventually

00:08:35.410 --> 00:08:38.150
to the result that
we want to have.

00:08:38.150 --> 00:08:40.420
So let's write out the binomial
coefficient and see if we

00:08:40.420 --> 00:08:41.780
can do something there.

00:08:41.780 --> 00:08:43.480
Oh, look at this.

00:08:43.480 --> 00:08:45.330
My iPod wants to sync.

00:08:45.330 --> 00:08:47.120
Let me get rid of that.

00:08:47.120 --> 00:08:50.030
All right, so then,
where was I?

00:08:50.030 --> 00:08:52.250
OK, so then this is equal to--
I'm just going to write out

00:08:52.250 --> 00:08:53.520
the binomial coefficient.

00:08:53.520 --> 00:08:55.960
k equals 1 to n.

00:08:55.960 --> 00:09:03.750
k times-- this right here is
n factorial over k factorial

00:09:03.750 --> 00:09:07.610
over n minus k factorial.

00:09:07.610 --> 00:09:13.970
Times p to the k times 1
minus p to the n minus k.

00:09:13.970 --> 00:09:16.370
And here we can make a little
bit of a simplification

00:09:16.370 --> 00:09:21.100
because what's k
divided by k factorial?

00:09:21.100 --> 00:09:24.870
Maybe I could rewrite it a
different way. k factorial is k

00:09:24.870 --> 00:09:29.530
times k minus 1 times k minus
2, and so forth, all the

00:09:29.530 --> 00:09:31.010
way until you get to 1.

00:09:31.010 --> 00:09:33.180
This is k factorial.

00:09:33.180 --> 00:09:40.270
So k factorial could be written
as k times k minus 1 factorial.

00:09:40.270 --> 00:09:43.300
It's k times, and then the
number 1 smaller then k times

00:09:43.300 --> 00:09:45.330
all the numbers below it.

00:09:45.330 --> 00:09:46.000
So let me rewrite.

00:09:46.000 --> 00:09:52.640
So this could be rewritten as
k times k minus 1 factorial.

00:09:52.640 --> 00:09:54.880
And the whole reason why I
did that is so I can cancel

00:09:54.880 --> 00:09:57.100
this k out with that k.

00:09:57.100 --> 00:10:00.540
So if I cancel that out I
think this warrants rewriting

00:10:00.540 --> 00:10:01.490
the whole thing again.

00:10:01.490 --> 00:10:05.900
So now, I guess you could argue
simplified it to, it equals the

00:10:05.900 --> 00:10:15.250
sum from k is equal to 1 to n
of n factorial over k

00:10:15.250 --> 00:10:17.920
minus 1 factorial.

00:10:17.920 --> 00:10:25.400
Times n minus k factorial times
p to the k times 1 minus

00:10:25.400 --> 00:10:28.560
p to the n minus k.

00:10:28.560 --> 00:10:30.350
And let's do another
simplification.

00:10:30.350 --> 00:10:32.270
Now, what I want to do and
we kind of know where

00:10:32.270 --> 00:10:34.140
we're heading, right?

00:10:34.140 --> 00:10:36.540
This should simplify
to n times p.

00:10:36.540 --> 00:10:39.320
So let's see if we can factor
out an n times p and then let's

00:10:39.320 --> 00:10:40.930
see if we can turn everything
else into a 1, and then

00:10:40.930 --> 00:10:42.230
we would be done.

00:10:42.230 --> 00:10:45.640
So we could rewrite n factorial
using the same trick up here.

00:10:45.640 --> 00:10:51.210
n factorial can be rewritten as
n times n minus 1 factorial

00:10:51.210 --> 00:10:52.710
by the same logic.

00:10:52.710 --> 00:10:58.450
And then p to they k could
be rewritten as p times

00:10:58.450 --> 00:11:01.610
p to the k minus 1.

00:11:01.610 --> 00:11:05.680
And then we can factor out this
n and this p and we'll get it's

00:11:05.680 --> 00:11:12.190
equal to np times the sum from
k is equal to 1 to

00:11:12.190 --> 00:11:14.470
n of-- let's see.

00:11:14.470 --> 00:11:16.360
We factored that n and p out.

00:11:16.360 --> 00:11:23.910
n minus 1 factorial over k
minus one factorial times

00:11:23.910 --> 00:11:28.210
n minus k factorial.

00:11:28.210 --> 00:11:32.140
Times p to the k minus 1.

00:11:32.140 --> 00:11:33.160
That's not in the denominator.

00:11:33.160 --> 00:11:40.150
This is just a regular-- times
1 minus p to the n minus k.

00:11:40.150 --> 00:11:40.900
And we're close.

00:11:40.900 --> 00:11:43.500
Remember, we want the result
that the expected value of our

00:11:43.500 --> 00:11:45.930
variable, and that's what
we were doing before.

00:11:45.930 --> 00:11:50.550
That this should
be equal to this.

00:11:50.550 --> 00:11:53.090
So we'll be done if we can
just show that this whole

00:11:53.090 --> 00:11:54.295
thing here equals 1.

00:11:54.295 --> 00:11:57.970
And to do that I'll make a
simplifying substitution.

00:11:57.970 --> 00:12:01.380
Let me make the substitution--
I don't know-- let's say that

00:12:01.380 --> 00:12:05.990
a is equal to k minus 1.

00:12:05.990 --> 00:12:10.710
And that b is equal
to n minus 1.

00:12:10.710 --> 00:12:14.270
And then what would
n minus k equal to?

00:12:14.270 --> 00:12:14.580
Let's see.

00:12:14.580 --> 00:12:18.920
If a is equal to k minus 1
then a plus 1 is equal to k.

00:12:18.920 --> 00:12:22.890
And then here, b plus 1 is
equal to n, so then n minus

00:12:22.890 --> 00:12:28.320
k would be equal to this,
a plus 1 minus this.

00:12:28.320 --> 00:12:32.850
Minus b minus 1, these
would cancel out.

00:12:32.850 --> 00:12:34.720
Which would equal a minus b.

00:12:34.720 --> 00:12:36.340
And let's see if we
can simplify this.

00:12:36.340 --> 00:12:44.020
So this whole sum will then
turn into np times the sum

00:12:44.020 --> 00:12:48.510
from-- OK, when k is equal to
1, that's the same thing--

00:12:48.510 --> 00:12:50.760
when k is equal to 1,
what is a equal to?

00:12:50.760 --> 00:12:53.350
a is equal to 0.

00:12:53.350 --> 00:13:02.230
From a equal to 0 to-- now
when k is equal to n,

00:13:02.230 --> 00:13:03.890
what will a be equal to?

00:13:03.890 --> 00:13:07.350
If this is equal to n,
if k is n, then a is

00:13:07.350 --> 00:13:09.830
equal to n minus 1.

00:13:09.830 --> 00:13:13.110
So we have a equal to a
to a equals n minus 1.

00:13:13.110 --> 00:13:15.760
But n minus 1 is the
same thing as b.

00:13:15.760 --> 00:13:17.870
So we could rewrite
the sum there.

00:13:17.870 --> 00:13:19.120
That's always a
little confusing.

00:13:19.120 --> 00:13:21.140
You might want to pause and
think about that a little bit.

00:13:21.140 --> 00:13:23.210
But I realize I'm already
over time, so I'll just

00:13:23.210 --> 00:13:24.970
keep moving forward.

00:13:24.970 --> 00:13:28.280
And then we had b is
equal to n minus 1.

00:13:28.280 --> 00:13:33.280
So that'll be b factorial
over k minus 1 , we made

00:13:33.280 --> 00:13:34.910
the definition that
that's equal to a.

00:13:34.910 --> 00:13:37.060
So that's a factorial.

00:13:37.060 --> 00:13:47.030
And then over here, n
minus k should be a--

00:13:47.030 --> 00:13:47.560
oh, you know what?

00:13:47.560 --> 00:13:50.750
I reversed this. n minus
ok should be b minus a.

00:13:50.750 --> 00:13:54.280
n minus k-- right.

00:13:54.280 --> 00:13:59.490
n is b plus 1, so it's b
plus 1 minus a plus 1.

00:13:59.490 --> 00:14:01.550
Minus a, minus 1.

00:14:01.550 --> 00:14:03.760
So the 1's cancel out
and you get b minus a.

00:14:03.760 --> 00:14:09.730
So the n minus k will become
b minus a factorial.

00:14:09.730 --> 00:14:14.580
And then p to the k minus 1--
k minus 1 is p to the a.

00:14:14.580 --> 00:14:19.230
And then times 1 minus
p to the n minus k.

00:14:19.230 --> 00:14:20.815
We already showed that
n minus k is the same

00:14:20.815 --> 00:14:23.510
things as b minus a.

00:14:23.510 --> 00:14:28.080
And then here, and we're
pretty much done now-- this

00:14:28.080 --> 00:14:31.760
right here, what is this?

00:14:31.760 --> 00:14:34.500
This is the probability
of-- well, let me rewrite

00:14:34.500 --> 00:14:35.320
it a simpler way.

00:14:35.320 --> 00:14:42.510
This is equal to np times the
sum from a is equal to 0 to b.

00:14:42.510 --> 00:14:43.080
What is this?

00:14:43.080 --> 00:14:44.816
This is b choose a.

00:14:44.816 --> 00:14:48.120
I have b things and I want to
choose a things from them, how

00:14:48.120 --> 00:14:52.580
many different ways can I--
times p to the a times 1

00:14:52.580 --> 00:14:56.220
minus p to the b minus a.

00:14:56.220 --> 00:14:59.380
What is this thing here?

00:14:59.380 --> 00:15:01.520
This is you're taking every
term of the binomial

00:15:01.520 --> 00:15:02.110
distribution.

00:15:02.110 --> 00:15:04.280
So you're saying, what
is the probability

00:15:04.280 --> 00:15:05.840
when a is equal to 0?

00:15:05.840 --> 00:15:08.990
This is the probability for
each of the a's, right?

00:15:08.990 --> 00:15:12.750
And you're summing up over all
of the a's you can achieve.

00:15:12.750 --> 00:15:17.360
So if I were to draw a quick
dirty distribution like this,

00:15:17.360 --> 00:15:19.700
if a is equal to 0 you have
a certain probability.

00:15:19.700 --> 00:15:22.280
Then a certain probability for
a equals 1. and then another

00:15:22.280 --> 00:15:23.490
probability, and
it goes higher.

00:15:23.490 --> 00:15:27.820
Then it's like a bell curve,
something like that.

00:15:27.820 --> 00:15:31.550
This term right here
is each of these.

00:15:31.550 --> 00:15:33.610
Each of these boxes you
could say represents

00:15:33.610 --> 00:15:34.740
one of these terms.

00:15:34.740 --> 00:15:36.460
When a equals 0 it's this term.

00:15:36.460 --> 00:15:38.240
When a equals 1 it's this term.

00:15:38.240 --> 00:15:41.580
When a equals 2 it's this
term, all the way to b terms.

00:15:41.580 --> 00:15:44.320
But we're summing them
up, so we're summing all

00:15:44.320 --> 00:15:45.860
of the probabilities.

00:15:45.860 --> 00:15:49.670
We're summing over all
of the values that our

00:15:49.670 --> 00:15:51.240
random variable can take.

00:15:51.240 --> 00:15:54.600
So if we solved all the
probabilities that a random

00:15:54.600 --> 00:15:58.040
variable can take, or we're
summing over all of the values,

00:15:58.040 --> 00:16:01.170
this is going to sum up to 1.

00:16:01.170 --> 00:16:04.420
This is like saying that this
is the probability of heads

00:16:04.420 --> 00:16:05.810
plus the probability of tails.

00:16:05.810 --> 00:16:08.630
Or you could say in the
flipping of a coin analogy,

00:16:08.630 --> 00:16:11.210
this is the probability that I
have one heads plus the

00:16:11.210 --> 00:16:13.210
probability I have 2 heads,
plus the probability I have 3

00:16:13.210 --> 00:16:15.465
heads, plus the probability I
have 4 heads, all the way to

00:16:15.465 --> 00:16:16.660
the probability I have b heads.

00:16:16.660 --> 00:16:19.690
So it's pretty much every
circumstance that can occur.

00:16:19.690 --> 00:16:22.640
So this is the sum over
the entire probability

00:16:22.640 --> 00:16:27.840
distribution, and so that
is going to be equal to 1.

00:16:27.840 --> 00:16:32.680
And so, we're left with
the expected value of our

00:16:32.680 --> 00:16:36.500
random variable, X, is
equal to n times p.

00:16:36.500 --> 00:16:39.390
Where n is the number of trials
and p is the probability

00:16:39.390 --> 00:16:40.830
of success on each trial.

00:16:40.830 --> 00:16:43.745
And this is true only for
binomial distributions.

00:16:43.745 --> 00:16:45.700
It isn't true for any
random variable, X.

00:16:45.700 --> 00:16:48.540
Only true for random variable,
X, whose probability

00:16:48.540 --> 00:16:51.230
distribution is the
binomial distribution.

00:16:51.230 --> 00:16:52.770
Anyway, I'm all out of time.

00:16:52.770 --> 00:16:54.910
I'll see you in the next video.