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# I How to bridge the gap between these approaches to SR?

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1. Aug 28, 2016

### leo.

For quite a long time now I'm having some trouble to bridge the gap between two different approaches to Special Relativity.

The first approach is the traditional one. It is the approach that Einstein presented in his paper and that is taught in most of the basic textbooks. In this approach, Special Relativity is introduced by two means of two postulates. In that setting the theory is based on the two postulates (the relativity principle and the constancy of the speed of light) and the Lorentz transformations arise as the transformations between inertial reference frames which obeys the two postulates.

In this approach, for example, it is the postulates that forces us to consider that time may be reference frame dependent, because the relativity of simultaneity is a consequence of the postulates alone.

On the other hand there is the modern approach. In the modern approach Special Relativity is introduced directly as a theory of the structure of spacetime. The spacetime metric is introduced right away and the Lorentz transformations are defined as the transformations of spacetime which keep the spacetime metric invariant.

On this approach, nothing leads to a new spacetime structure, nor to the metric. It all appears out of thin air, inasmuch as the postulates in the original approach.

Now I want to make something clear here. I do know that the only important thing is that both approaches work. One can look into the second approach, and just use it.

The point is that IMHO, there's a huge gap between the two approaches.

The original approach by Einstein is well motivated in his paper. Talking about Electrodynamics, Einstein is able to motivate quite well the need for his two postulates. It all flows quite smoothly.

The second approach is not that nice to grasp. It doesn't seem to flow smoothly from anything else. It is usually not motivated, it is just presented as: "the mathematical framework is this because it works" and nothing more is said.

Furthremore, it is not at all obvious that the Einstein's postulates are totally equivalent to invariance of the spacetime metric.

In that sense, how can we bridge the gap between the two approaches? How can we present the second approach in a quite smooth and natural way as the first approach was presented by Einstein?

2. Aug 28, 2016

### QuantumQuest

The key here, is the kind of geometry that is used. It is hyperbolic, so the metric is according to this. On the other hand, the postulates of Einstein lead to this kind of geometry too.

3. Aug 28, 2016

### Staff: Mentor

Einstein's postulates imply the Lorentz transforms, and the Lorentz transforms imply the invariance of the spacetime interval. Is that not sufficient?

Einstein's approach only seems smooth and natural because, with the advantage of hindsight, we're able to avoid a number of wrong turns and unnecessary detours.
- Relativistic mass accepted then discarded.
- Applicability of SR to accelerated coordinate systems missed the first time around.
- Value of Minkowski model only appreciated after the fact.

The historical approach got to us the destination, but now that we know what the destination is, we can find easier paths to it. Once you know that the Minkowski spacetime geometry ensures the invariance of the speed of light, it's not unnatural to take the assumption that the universe is Minkowskian as a starting point equivalent to Einstein's postulates. Of course it would never occur to anyone to try that unprompted if they hadn't already seen the historically derived answer - but that's what hindsight is for.

4. Aug 28, 2016

### robphy

5. Aug 29, 2016

### strangerep

Heh, well, I'm from the camp that thinks the 2nd approach is "cheating".

Mathematicians tend to prefer the 2nd approach because it's more axiomatic, whereas the 1st approach is more physically motivated.

Once you have the symmetry group (i.e., Poincare transformations), the spacetime can be constructed as a homogeneous space for the group. (This is in fact an aspect of the Klein's Erlangen program, which featured briefly in a recent Insights article.)

So I wouldn't even bother starting from Minkowski spacetime, but that's just my personal taste.

BTW, there's actually a "zero'th" approach which starts from the Relativity Principle, but does not assume the Light Principle. I.e., it doesn't assume invariance of the speed of light. A sketch of this approach appears in this Wiki page (section "From group postulates"). A full exposition of this derivation is rather long, however.

6. Aug 29, 2016

### vanhees71

It depends on whom you address to teach SR. As a first encounter, I think the Einstein way is the best. You start with the tension between the classical (Newtonian) space-time structure and electromagnetism by the observation that Maxwell's equation are not Galileo-invariant. Einstein's ingenious insight (after many years of thinking about the problem!) was to keep precisely the essence of what's right about both theories: From Newtonian mechanics he took the principle of inertia and the corresponding existence of a class of preferred reference frames, namely the inertial frames (even in GR this holds with the qualification that it becomes a local concept, and locality should be emphasized in relativistic physics anyway!). Then from Maxwell's equations and the assumption that they should be forminvariant under transformations from one inertial reference frame to another he concludes that the only thing which goes wrong in using the Galileo transformation is that the speed of light wouldn't be the same. So as the essence of getting both the principle of special relativity and Maxwell's equations consistence is to postulate the constancy of the speed of light. No matter how the light source moves (with constant velocity wrt. to an inertial frame of reference!) the speed of light (i.e., the phase velocity of electromagnetic waves in a vacuum) should be the same. From this he derives the Lorentz transformations. From this it is only a very small step to analyze the correct mathematical frame work, i.e., Minkowski geometry (space time=pseudo-Euclidean affine manifold with a pseudo-metrix of signature (1,3)).

The "axiomatic approach" is a shortcut for advanced students, starting right away from the Minkwoski space is a shortcut emphasizing the math rather than the physics.

There's a 3rd approach, which however is a bit lengthy compared to the other two, but for me provide the most mathematical fun: You just postulate the principle of special relativity, the homogeneity of space and time (translation invariance) and isotropy of space for any inertial observer and assume that the space-time symmetry transformations build a continuously connected group. This leads to precisely two spacetime structures. You can guess easily, which two that might be ;-)).

7. Aug 29, 2016

### micromass

Do you have any good reference on this?

8. Aug 29, 2016

### vanhees71

9. Aug 29, 2016

### soviet1100

I've only recently studied general relativity, so I might not be as qualified as the others to make a properly informed statement, but I'll add my own two cents.

I was first taught SR using lights and clocks and trains and paradoxes and Maxwell equations. It didn't work well and it made ugly one of the most, if not the most, subject in all of physics. Same goes for electrodynamics taught starting from Coulomb's law and biot-savart's law etc. and subsequently fixing one stupid mistake after the other (this is the approach followed in for e.g. Griffiths)

The approach that I would vehemently (militantly?) advocate is that taken in Landau & Lifshitz's CFT (Vol. 2). First postulate the existence of a maximal velocity and the equivalence of inertial frames. These postulates aren't at all like the postulates of QM in that these are easily digestible and there exists a wealth of experimental support that one can easily find in papers and understand. Once you do that, you can systematically reason/derive the invariant interval between events. L&L do this on page 4; I'm not reproducing their argument here. This essentially fixes the 'metric', if you want to call it that. But you can argue the case for the invariant interval from purely physical perspectives. Once that's done, the only other "assumption" is the holy grail of physics, the principle of extremal action. The equivalence of inertial frames postulate imposes strong constraints (lorentz invariance) on possible terms in lagrangians. From these basic ingredients, one can derive all of SR (including Lorentz transformations, concepts of simultaneity etc.), E&M, and then beautifully generalise to GR. When this approach is used in GR, the beauty is breathtakingly stunning. Without mention of the mathematical formalism of manifolds or charts or covariant derivatives or geodesics, all these concepts just plumply fall into your lap from just the action principle. It's actually a great platform to intuitively understand some basic concepts of diff geo. It's a shame that the action principle is not emphasised enough, if at all, in undergraduate university courses.

{begin rant} This is how physics is supposed to be taught, from scratch. Starting with simple comprehensible experimental data, translating them into mathematical requirements of a theory, then choosing a mathematical structure, building it, and then finally making testable predictions with it. Violations against this is most prominent in teaching QM, where everyone usually just postulates out of the blue that a system is described by a vector in a Hilbert space with dynamics governed by the Schrodinger equation etc. And the reason why such is the case is usually summed in a single statement: "because it ultimately agrees with experiment". I doubt someone looked at an electron diffraction apparatus and then straight away concluded that the electron lives in a hilbert space and wrote down the Schrodinger equation. One book that does justice is of course Dirac's book (read in conjunction with Ballentine's chapter 3) {end rant}.

Last edited: Aug 29, 2016
10. Aug 30, 2016

### vanhees71

I couldn't agree more with what you said about relativity and electrodynamics. I don't understand why they copy still 19th-century textbooks concerning E&M. I think the final word on this kind of textbooks has been written by Jackson and that's it. For the 21st century, LL is the right way, although one needs to also provide a step between the inductive way the subject is treated in the experimental-physics course and theory, including a thorough treatment of vector calculus, which is the main obstacle for undergraduate students to learn the subject.

Concerning QT, I'd say you need a bit of historical introduction to motivate the Hilbert-space "axiomatics". It's too abstract to just being thrown in in the first lecture. That doesn't help to gain understanding, why this abstraction is necessary. I think the best approach is the heuristics via de Broglie-Einstein ("wave particle dualism") with the clear hint that this is not satisfactory and then coming as quickly as possible to Dirac's representation free formalism with abstract operators and vectors on Hilbert space (and no, the pure states are not represented by the vectors but by rays!).

11. Aug 30, 2016

### leo.

Thanks for all the answers. I agree that the first approach is more physically motivated and easier to understand from a physical standpoint. On the other hand I'm starting to think that the best way to motivate the second approach would then to see historically how people got there.

And this is indeed the sort of "gap" that I have the impression that exists between the two approaches. I mean, in most of the presentations I've seem up to now things goes more or less along these lines:
1. Einstein's original approach is presented (or just a recap of it), deriving the relativity of simultaneity, length contraction, time dilation, and hence, the Lorentz transformations (in truth just the boosts).
2. Without any motivation the spacetime metric (and hence the metric spacetime interval) is defined and it is shown to be able to classify the separation of events (timelike, spacelike or lightlike).
3. The spacetime interval is shown to be invariant under boosts.
4. It is immediately said that the Lorentz transformations can be just characterized directly as the transformations preserving the spacetime interval.
I mean, as one goes from Classical Mechanics and Electrodynamics to Einstein's approach is, as I said, smooth and natural. Following Einstein's approach is also going in a quite natural direction.

I feel that the gap appears in two distinct situations: when the metric tensor is presented (which is at first a quite strange thing to define and usually has no motivation presented) and later when it is said that the spacetime metric is enough to rebuild the entire theory.

I mean, that the spacetime metric is enough to rebuild the entire theory is not at all obvious. If we look at it carefully, it seems to be just another object defined in the theory that happens to be invariant under boosts. I can't see how it is obvious that the other way around (that the Lorentz transformations can be characterized as the ones keeping the metric fixed) works and rebuilds the theory.

Is there one way to fill those gaps? I mean, is it possible to:
1. Motivate the introduction of the spacetime metric properly, so that it doesn't appear to just come out of thin air.
2. Motivate and make it quite reasonable that defining the Lorentz transformations as the ones preserving the spacetime metric we get the same transformations arising from Einstein's postulates and we are able to rebuild the entire theory.
Of course, as I said, this motivation might be historical, might be just to say how historically people did this for the first time. Perhaps the only possible way to do it is to explain historically what was done, but I never found this data.

12. Aug 30, 2016

### robphy

Did you see my reference to the Bondi k-calculus in #4?
There the metric and the Lorentz transformation are not in the foreground of the discussion.
It is the principles of relativity, with focus on the radar method and the Doppler effect.
(Secretly, the approach is using the eigenbasis of the Lorentz Transformation.)

https://archive.org/details/RelativityCommonSense
https://en.wikipedia.org/wiki/Bondi_k-calculus

13. Aug 30, 2016

### vanhees71

How do you like my PF-FAQ article?

http://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

There I use a mixture of Einstein's postulates and the argument via the pseudo-Euclidean affine structure of spacetime (it's in fact not a metric but a pseudometric in the sense of a pseudo-Riemannian manifold, of which Minkowski space, i.e., the pseudo-Euclidean affine space with signature (1,3)).

14. Aug 30, 2016

### samalkhaiat

15. Aug 31, 2016

### Battlemage!

I saw one book where it focused on general linear transformations, then it noted (if I recall correctly) that the Galilean transformation (with c=infinity) and Lorentz transformation are the only possible linear transformations that could work with the assumption of the relativity principle.

I think a fun approach would be to start with some general transformation, for example you might have x coordinates transform by x = f(ax +bt), then show that the Galilean transformation is a special case of it, and then show that the Lorentz transformation is as well; and finally, show that the Galilean transformation is itself a special case of the Lorentz transformation.

At least with that approach you'd touch a little bit on the math structure, while not sacrificing the central component: the principle of relativity.

16. Aug 31, 2016

### robphy

...but that's not a true statement.
One could argue that a Galilean boost is a certain limit (but not a special case) of a Lorentz boost.

17. Aug 31, 2016

### Battlemage!

How so? Aren't all limits a special case of a particular function? Anyway, maybe I should update my book collection. This entire idea came from Rindler's Essential Relativity, which says on page 65 "Thus we are left with LT's having arbitrary positive "lightsquare" V2, and GT's. The latter are really particular cases of the former (corresponding to V2 = ∞) [...]", so just fyi I didn't just make that up.

Anyway, a limit that is what I meant. You could write the Galilean transformation identically with gamma as 1/sqrt(1-(v^2/c^2)) and then write "with c->infinity" afterwards and have the same thing. Kind of like saying some constant c is a polynomial but with x=0. I mean, P(x) = c for all x in R, c = constant.

So, my point is, you can have a generic coordinate transformations: x = ƒ(v)*(ax + bt), t = g(v)*(et + fx), and then simply replace the coefficients and/or functions with what is needed to get both the GT and LT.

My retelling of it, due to my lack of experience, was technically incorrect, however as I said I got the idea from a pretty well known book: an old version of Rindler's Essential Relativity, pages 64-67 and . It considered what SR would look like without the c speed limit postulate, and then went super abstract with generic transformations, pointing out that the principle of relativity is only preserved with a GT or LT transformation (and pointing out the differences in between universes with each one).

So ignoring the technical failures of yours truly, I still like the idea of approaching it from a "top down" view instead of inexplicably (in the eyes of the student) jumping from GT's to LT's. The first time I saw the GT equations I thought, "well that makes sense." First time I saw the LT equations I was like "wat..."

18. Aug 31, 2016

### robphy

My answer to "Aren't all limits a special case of a particular function?" is NO.
In 1+1 Minkowski spacetime, a lorentz boost (in rectangular coordinates with natural units) is symmetric... a Galilean boost is not.

Yes, when you write the Lorentz Transformations in a certain way and then take the limit of this "maximum signal speed parameter" as it goes to infinity,
you can get to the Galilean limit. (If you write it a different way, you get the so-called [Lewis] Carroll limit. So, what you write down [to describe what limit is being taken] before taking the limit is important.)
These are discussed in http://scitation.aip.org/content/aip/journal/jmp/9/10/10.1063/1.1664490
"Possible Kinematics" - Henri Bacry and Jean-Marc Lévy-Leblond
J. Math. Phys. 9, 1605 (1968)

https://arxiv.org/abs/1402.0657 "Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time"

With this regard, I also advocate a top-down approach... this approach is based upon the Cayley-Klein Geometries (which likely includes [subsumes] these "relativity without the speed of light postulate"... including Rindler, Mermin (1984) http://scitation.aip.org/content/aapt/journal/ajp/52/2/10.1119/1.13917 and Ignatowski (1910) https://en.wikisource.org/wiki/Translation:Some_General_Remarks_on_the_Relativity_Principle .)

Three of them are well known to mathematicians (Euclidean, Elliptic, and Hyperbolic), the other six are three Lorentz-signature spacetimes (Minkowski, deSitter, and anti deSitter) and their Galilean limits.

In this scheme, one can write a "unified", "generalized", (or "meta"-)expression for a "boost"... of which Minkowski, Galilean, Euclidean, deSitter, etc... are special but distinct (non-overlapping) cases. The appropriate "experiment" can then be used to pick out the case under study.

Last edited: Sep 2, 2016
19. Aug 31, 2016

### Battlemage!

I just wonder if there is a way to teach it this way without needing the student to understand more than the basics of linear algebra, because by the time a student usually starts getting into these types of non-euclidean geometries, they've already been acquainted with the basics of special relativity.

As an undergrad I have barely seen non-euclidean geometry, but I saw the basics of SR my first year. Of course it was inexplicable to me from a general math perspective. It made sense to talk about c as a speed limit and then derive the transformations from that, but at the time the only explanation was "well, we observe c=constant therefore SR." And then I've been told later on in GR that isn't strictly speaking true, since the notion of a speed is not easily defined in the first place. I just don't know if GR is where a student should be introduced to non-euclidean geometry. It's like saying "well, we kind of lied to you before but here is how things really are..."

Then again maybe that is just too much that early.

20. Aug 31, 2016

### robphy

One of my ideas is to start by slowly developing aspects of the non-euclidean geometry already inherent in a PHY 101 position vs time graph.
Although the simulation already starts at the "E=1" case [the punchline],

(Based on an earlier post of mine https://www.physicsforums.com/threads/minkowski-diagram.876359/#post-5503686 )

https://www.desmos.com/calculator/ti58l2sair [my t axis is horizontal ]

The Lorentz Transformations mentioned above preserves the unit hyperbola (that is, under that Lorentz Transformation, the events on the future unit hyperbola are mapped to the same hyperbola), whose asymptotes are lightlike---that is to say, all frames of reference agree on the same value of the speed of light (maximum signal speed).

An inertial worldline (the t'-axis) drawn from the "center" of the hyperbola into the future light cone intersects the hyperbola at an event.
Think of this worldline as a radius vector from the center.
The tangent line to the hyperbola here is "spacetime-perpendicular" (perpendicular in Minkowski spacetime geometry) to the worldline.
Think "tangent to a circle is perpendicular to radius".
Physically, this tangent line determines the x'-axis.... That observer's "space" direction is perpendicular to that observer's "time" direction.
Additionally, this tangent line has the "same value of t' " for that observer---this is that observer's line of simultaneity.

Try changing the E-slider.... to the Euclidean case (when E= -1) and the Galilean case (when E=0).


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13 is a linear transformation and .......Determine T

• MHB
Gold Member
MHB
Suppose that $T: \Bbb{R}^3 \rightarrow \Bbb{R}^3$ is a linear transformation and
$$T \begin{bmatrix} 1 \\1 \\0 \\ \end{bmatrix} = \begin{bmatrix} 1 \\2 \\1 \\ \end{bmatrix}, \quad T \begin{bmatrix} 1 \\0 \\1 \\ \end{bmatrix} = \begin{bmatrix} 1 \\0 \\2 \\ \end{bmatrix}, \quad T \begin{bmatrix} 0 \\1 \\0 \\ \end{bmatrix} = \begin{bmatrix} 2 \\2 \\3 \\ \end{bmatrix}.$$
Determine $T \begin{bmatrix} 1 \\2 \\3 \\ \end{bmatrix}$

ok this should be easy... but.. the examples were not that close to this

Gold Member
MHB
Suppose that $T: \Bbb{R}^3 \rightarrow \Bbb{R}^3$ is a linear transformation and
$$T \begin{bmatrix} 1 \\1 \\0 \\ \end{bmatrix} = \begin{bmatrix} 1 \\2 \\1 \\ \end{bmatrix}, \quad T \begin{bmatrix} 1 \\0 \\1 \\ \end{bmatrix} = \begin{bmatrix} 1 \\0 \\2 \\ \end{bmatrix}, \quad T \begin{bmatrix} 0 \\1 \\0 \\ \end{bmatrix} = \begin{bmatrix} 2 \\2 \\3 \\ \end{bmatrix}.$$
Determine $T \begin{bmatrix} 1 \\2 \\3 \\ \end{bmatrix}$

ok this should be easy... but.. the examples were not that close to this
Well, the most direct method would be to simply solve for T. But since T is linear there is another way.

Can you build $$\displaystyle \left [ \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right ]$$ out of a linear combination of $$\displaystyle \left [ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right ]$$, $$\displaystyle \left [ \begin{matrix} 1 \\ 0 \\ 1 \end{matrix} \right ]$$, and $$\displaystyle \left [ \begin{matrix} 0 \\ 1 \\ 0 \end{matrix} \right ]$$?

-Dan

Gold Member
MHB
ok this probably is not exactly what it is supposed to be but

$T \begin{bmatrix} 1 \\1 \\0 \end{bmatrix} =\begin{bmatrix} 1 \\2 \\1 \end{bmatrix}, \quad T \begin{bmatrix} 1 \\0 \\1 \end{bmatrix} = \begin{bmatrix} 1 \\0 \\2 \end{bmatrix}' \quad T \begin{bmatrix} 0 \\1 \\0 \end{bmatrix} = \begin{bmatrix} 2 \\2 \\3 \end{bmatrix}$.

$=T\left[\begin{array}{c} 1&1&0\\ 1&0&1\\ 0&1&0 \end{array}\right]$

or possibly
$\left[\begin{array}{c}x_1+x_2 \\x_1+x_3\\x_2\end{array}\right]$

Last edited:
Gold Member
MHB
ok this probably is not exactly what it is supposed to be but

$T \begin{bmatrix} 1 \\1 \\0 \end{bmatrix} =\begin{bmatrix} 1 \\2 \\1 \end{bmatrix}, \quad T \begin{bmatrix} 1 \\0 \\1 \end{bmatrix} = \begin{bmatrix} 1 \\0 \\2 \end{bmatrix}' \quad T \begin{bmatrix} 0 \\1 \\0 \end{bmatrix} = \begin{bmatrix} 2 \\2 \\3 \end{bmatrix}$.

$=T\left[\begin{array}{c} 1&1&0\\ 1&0&1\\ 0&1&0 \end{array}\right]$

are we trying to build $Ax=B$
What I am trying to point you toward is
$$\displaystyle \left [ \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right ] = a \left [ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right ] + b \left [ \begin{matrix} 1 \\ 0 \\ 1 \end{matrix} \right ] + c \left [ \begin{matrix} 0 \\ 1 \\ 0 \end{matrix} \right ]$$

so
$$\displaystyle T \left [ \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right ] = a T \left [ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right ] + b T \left [ \begin{matrix} 1 \\ 0 \\ 1 \end{matrix} \right ] + c T \left [ \begin{matrix} 0 \\ 1 \\ 0 \end{matrix} \right ]$$

Note that this can only be done if T is linear.

-Dan

Gold Member
MHB
then?
$\left[\begin{array}{c}Ta+Tb \\Ta+Tc\\Tb\end{array}\right]$

i think I am getting confused by looking at too many examples

how would we know if T is linear?

Gold Member
MHB
how would we know if T is linear?
Suppose that $T: \Bbb{R}^3 \rightarrow \Bbb{R}^3$ is a linear transformation

then?
$$\displaystyle T \left [ \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right ] = a T \left [ \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right ] + b T \left [ \begin{matrix} 1 \\ 0 \\ 1 \end{matrix} \right ] + c T \left [ \begin{matrix} 0 \\ 1 \\ 0 \end{matrix} \right ]$$
and
$$T \begin{bmatrix} 1 \\1 \\0 \\ \end{bmatrix} = \begin{bmatrix} 1 \\2 \\1 \\ \end{bmatrix}, \quad T \begin{bmatrix} 1 \\0 \\1 \\ \end{bmatrix} = \begin{bmatrix} 1 \\0 \\2 \\ \end{bmatrix}, \quad T \begin{bmatrix} 0 \\1 \\0 \\ \end{bmatrix} = \begin{bmatrix} 2 \\2 \\3 \\ \end{bmatrix}.$$

As for
$\left[\begin{array}{c}Ta+Tb \\Ta+Tc\\Tb\end{array}\right]$
$Ta$ does not makes sense for $a\in\mathbb{R}$ because $T:\mathbb{R}^3\to\mathbb{R}^3$.

Gold Member
MHB
ok apparently I'm not understanding the steps
not sure what I should be asking

format: plain


text:
# Probability outcomes for coins, dice, and spinners

#### All in One Place

Everything you need for better grades in university, high school and elementary.

#### Learn with Ease

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0/2
##### Intros
###### Lessons
1. Introduction to Probability Outcomes for Coins, Dice, and Spinners:
2. What is probability?
3. What are outcomes in probability?
1. Outcomes for flipping a coin
2. Outcomes for rolling a 6-sided die
3. Outcomes for a spinner with different regions
0/7
##### Examples
###### Lessons
1. Outcomes are the result of flipping coins, rolling dice, or turning a spinner's arrow
List all the possible outcomes. How many outcomes are there in total?
1. A coin with a happy face or sad face on its sides
2. An 8-sided die
3. Two friends having a race against each other
2. Outcomes that are equally as likely to occur
Fill in the table for the probability outcomes
1. Rolling a 6-sided die:

2. Turning a spinner with letters:

3. Turning a spinner with numbers:

4. Rolling a 6-sided die:

###### Topic Notes

In this lesson, we will learn:

• What is probability in math?
• How to write probability as a fraction
• How to list all the outcomes for probability events (using coins, dice, and spinners)

Notes:

• Probability is math for events that only sometimes happen. The chances of something happening can be likely or unlikely to happen.
• Calculating probability is like predicting the future. We are trying to get a measure of the chances that something will happen.

• Arithmetic math (all basic operations; adding, subtracting, multiplying, dividing numbers) is certain. In comparison, probability math is uncertain--but still predictable.

• Probability can be given as a fraction, following the formula format:

• Probability = $\frac{number\,of\, outcomes\, wanted} {total\,number\,possible\,outcomes}$

• Outcomes are all the possible endings that could happen for a situation.
• Some simple probability situations that are often used in math problems include: tossing a coin, rolling a six-sided die, and spinning the arrow on a spinner.

• Coins:
• If you toss a coin, it will land on a flat side—either on heads or tails
• So, there are 2 possible outcomes to a coin toss: heads or tails
• The probability of landing on heads is $P$(heads) = $\frac{1}{2}$
• The probability of landing on heads is $P$(tails) = $\frac{1}{2}$

• Dice: (a six-sided die, if singular)
• When rolling a die, it will land with one of its six flat faces facing up (on top). There are six sides labelled from 1 to 6.
• So, there are 6 possible outcomes to a die roll: 1, 2, 3, 4, 5, or 6
• The probability of landing on any one side (1-6) is one out of six chances
• $P$ (1) =$P$ (2) = $P$ (3) = $P$ (4) = $P$ (5) = $P$ (6) = $\frac{1}{6}$

• Spinners: (can have any number of equal parts; labels with numbers, letters, etc.)
• When using a spinner, the arrow will land on one of the marked regions. In this case, there are 4 different coloured regions.
• So, there are 4 possible outcomes for this spinner: red, yellow, green, or blue
• The probability of landing on any one of the colors is one out of four chances
• $P$ (Red) = $P$ (Yellow) = $P$ (Green) = $P$ (Blue) = $\frac{1}{4}$


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# E[x|y] where y=cdf(x) and y is random

I have a grid approximation of a cdf, $F_x$. The cdf has support for $x>=0$

From there, calculating the $E[x]$ is straight forward with some std numerical integration techniques.

In my case, however, I have a variable $y$, which is randomly distributed according to a pdf $f_y$.

$y$ defines the percentile in which $x$ is likely to be. The relationship is $y = F_x(x)$

My goal is to calculate the $E[x]$ by knowing that $y$ is distributed as $f_y$. Can I just simply calculate the $E[x]$ integrating with $f_y(x)$ mapping $y$ to $x$ using $x = F_x(y)^{-1}$ (the quantile function). Or is there something that gets lost there (which is my concern)?

EDIT

I have been reflecting on the problem formulation.

In a Bayesian framework where $c$ represents client data and $s$ represents sales

we have:

$$p(s|c) = \frac{p(c|s) * p(s)}{p(c)}$$

In the current set up: $p(s)$ is an empirical $cdf$ - which looks very weird and I would not try to fit - while $c$ is a ranking on clients. I was wondering if in this scenario we get to a numerical estimate of $E[s|c]$ and what is the role played by the probability integral transform ?

• That's fine. In general, if $y$ has some distribution and $x$ is some function of $y$, $h(y)=x$ then $E[x] = \int h(y) f_y dy$. This is the so called law of the unconscious statistician! – CloseToC Jun 10 '15 at 17:55
• If $$Y = F_X(X)$$ then by the probability integral transform $y$ is necessarily distributed as a $U(0,1)$ Uniform, not according to some arbitrary distribution. – Alecos Papadopoulos Jun 10 '15 at 18:07
• @Alecos: This is true. This is the missing piece. Tnx. But in my current setup I have, in some cases, information on the ranking, the distribution on $y$. How to incorporate that? – IcannotFixThis Jun 10 '15 at 20:06
• You mean you have information on realizations of $Y$? – Alecos Papadopoulos Jun 10 '15 at 20:07
• Unfortunately, you are, because the distribution of $Y$ if $Y$ is defined as above, is always $U(0,1)$, no matter what any estimation of it based on any finite sample may show. – Alecos Papadopoulos Jun 10 '15 at 22:01


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# Using Backup and Migrate Module to download windows compatible backup

I have created a Drupal7 based web-site on my hosting server. I have made changes to the themes, created views and done other customizations. I would like to create a duplicate copy of the site on my local laptop, so that I can test my changes locally before making them live on the site. I installed the Backup and Migrate module on my host server and downloaded a backup of the entire site to my windows 7 laptop. I already have a WAMP installation and all required services running for creating drupal7 site.

The file I downloaded looks like a tar ball of the entire drupal folder and the DB and the size is 106 MB. Since, it is a tar ball I could not open the existing file in windows. My questions:

1. Is there a way to get around the tar ball? I tried specifying zip as the compression format, but that did not help. I do not see any option to specify no tar.
2. What is the correct way to create a copy of the site locally, considering I have made changes not only to DB, but other files too. I have seen other options such as exporting the DB and FTPing the sites folder. If I were to do this, how do I FTP the sites folder to my windows laptop?

Thanks

There's a couple steps you still need to do:

1. To "untar" the tar ball on Windows, you just need the right software for it. An easy to use and free software that supports TONS of compression types is 7-Zip -- http://www.filehippo.com/download_7-zip_64.

2. Once you have all the files available to you, you'll need to import the database into MySQL. Luckily WampServer comes with phpMyAdmin. You can find a link to it in the tray icon for WampServer. First create your database, then use the Import tab to load in the SQL file.

3. Modify the settings.php file to match the database database connection information for your local MySQL installation. The default installation of Drupal creates this file is at /sites/default/settings.php.

4. Then visit http:// my-local-host/ (or http:// my-local-host/[my-drupal-subdirectory]/) to see if it's working. (Replace "my-local-host" with "localhost" since StackExchange doesn't like "localhost" in a link).

• Thanks for the answer. When I create my database and import the sql file, I get an error saying column too long, use row_format=dynamic or compression. Is there a workaround to overcome this. – cppcoder Oct 2 '14 at 17:53
• I've been using MySQL for many years on many different hosts and never had that problem, so it sounds to me like a configuration setting on your remote server which is different than your local server. If you can get access to phpMyAdmin on your remote server, you can manually export to a file (don't forget to set the compression) and then try importing that locally. Otherwise, you might try changing your local MySQL server settings per the answer here: stackoverflow.com/questions/24321896/… – hargobind Oct 3 '14 at 7:10

I believe PHP supports output streams in gzip, tar, zip -- you simply need the correct PHP extension installed/enabled in your php environment. To my knowledge Backup and Migrate will make use of the available options in your configuration.

In the 2.x branch, zip is an allowed filter: http://cgit.drupalcode.org/backup_migrate/tree/includes/filters.compression.inc#n77

You can see what your php configuration is easily on a system using php_info() on the command line or looking at admin/status/reports in the drupal administration area under the PHP report.

What is the correct way to create a copy of the site locally, considering I have made changes not only to DB, but other files too. I have seen other options such as exporting the DB and FTPing the sites folder. If I were to do this, how do I FTP the sites folder to my windows laptop?

Backup and Migrate (BAM) is a fine way to backup a full drupal site. It has many contributed addon modules to backup to other destinations, such as Dropbox if needed. An additional option to BAM is Drush and the archive-dump command which gives you a .tar.gz (hehe) of an archive-manifest of your full site. You can the easily restore the site via commandline using drush archive-restore /path/to/dump.tar.gz .... see drush help archive-restore for full documentation examples.

I'm sorry I can't answer your question how to use FTP, this is a drupal only forum and general programming question should be resolved via Google or StackOverflow for instance.

• I found a software called 7-zip which helped me to untar and extract the tar ball. I'm trying to create a new test site and will let you guys know how it goes. Btw, I have been hearing about drush - will it provide a linux like command line in windows. – cppcoder Oct 1 '14 at 22:08
• i hear google is great at answering how to use drush. as is the github drush page. – tenken Oct 1 '14 at 22:17


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# How do we justify functions on the Ordinals

In ZFC there seem to be two ways to define a function, as an ordered triple of domain, codomain and the set of ordered pairs that make its graph, or a relation over the two sets.

Whichever we use doesn't matter, but this seems to present a certain problem. Both are defined over sets, but no-one seems to have any problem defining functions on the ordinals like Hartog's function giving the minimum ordinal that doesn't have an injective function on to the given ordinal. I know we want to be able to obviously, but how are we supposed to justify this, because the definitions of functions don't seem to include classes. Or is there a nicer definition I've missed?

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## 1 Answer

Instead of a set, require that a function is a collection of ordered pairs with the property that if $\langle u,x\rangle$ and $\langle u,y\rangle$ are both in the collection then $x=y$.

Classes in ZFC are syntactic objects, these are the definable sub-collections of "$V=\{x\mid x=x\}$", so a class is a collection (which might not be a set) of the form $\{x\mid\varphi(x)\}$ for some $\varphi$.

Ordinal functions, or in general class functions, are (in the context of ZF) definable classes of the form $\{\langle x,y\rangle\mid\varphi(x,y)\}$ which has the functional property. For example the Hartogs number function makes use of the fact that we can define from ZF what are the ordinals, what are injective functions, etc. then using a parameter $x$ we can define the least ordinal which is not injected into $x$. This goes into a formula $\varphi(x,y)$ which says all of the above.

Two simple and even more natural examples of class functions are:

1. The power set function, $\{\langle x,\mathcal P(x)\rangle\mid x\in V\}$ where $\mathcal P(x)$ denotes the power set of $x$. The axiom of power set tells us that this set exists and extensionality tells us it is unique.

2. The union function, $\{\langle x,\bigcup x\rangle\mid x\in V\}$ where $\bigcup x=\{z\mid\exists u(u\in x\land z\in u)\}$. Again we adhere to the axioms, this time to the axiom of union and the axiom of extensionality to argue that this function is well defined.

The von Neumann hierarchy, the Constructible hierarchy, the Fine structure hierarchy, and many other similar ways to construct models of set theory are exactly class-functions from the ordinals.

To read some more:

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This seems incredibly simple, I love it, thank you. I've gone through so many resources all using the definitions I gave, when they just don't work, it's a wonder they get published. Is there any chance you'd know of any references for this definition? While I'm happy with it, I don't think stackexchange is something I can reference. – AaronM Apr 26 '12 at 17:58
@AaronM: I suppose you are not publishing in a logic related journal, because I think this sort of thing is too simple to actually care about. I'll run through Jech and several other introductory books which have a few remarks about classes, <strike>if I will find something I will let you know.</strike> Jech, Set Theory pp. 11-12 (bottom of page 11, top of 12) he remarked that the definitions of a function can be applied to classes as well. I suppose that should do, or do you need to reference exactly how classes are treated in ZF? – Asaf Karagila Apr 26 '12 at 18:00
Haha I guess it shows I'm new to published stuff. Thank you very much though, this is a far better answer than I hoped for, you've been a brilliant help. Edit: That will do brilliantly thank you. I've had a hard time getting hold of that book, typical it would be the one which has the answer I want! :p – AaronM Apr 26 '12 at 18:06
@AaronM: I haven't published anything anywhere. However this sort of thing is not the type of things you see mentioned in papers beyond mentioning that a particular function is a class function, and justifying the well-definition of the formula defining the class if needed. – Asaf Karagila Apr 26 '12 at 18:11