## Game Show Problem

**Moderator:** Marilyn

### Re: Game Show Problem

Not nearly as interesting as hearing Jeff explain the phrase "The possible outcomes for one coin toss can be described by the sample space {Heads,Tails}.".

Note that it is the coin toss that is described by the sample space, and NOT, as Jeff wants it, that the sample space is described by {Heads,Tails}.

Note that it is the coin toss that is described by the sample space, and NOT, as Jeff wants it, that the sample space is described by {Heads,Tails}.

- Gofer
- Intellectual
**Posts:**199**Joined:**Mon May 09, 2016 8:24 am

### Re: Game Show Problem

And yet still not recognizing the words "described by" or "can be."

Note that it is the outcomes that are described by what, if we follow Gofer's preferred sources, the state space {Heads,Tails}. But since it is almost never necessary to be as pedantic as Gofer wants us to be, many people equate the two.

AGAIN: The sample space is the set of all possible outcomes. The model of the sample space that we use, properly called the state space if you want to be that pedantic, is the set of all possible combinations for the set of random variables we choose to use to describe it.

Isn't it interesting that Gofer never tries to rebut this? That he only asks me to explain ambiguous references that he cherry-picks to maybe, possibly, if you ignore enough definitions, justifies his position? Which he never clearly states?

Note that it is the outcomes that are described by what, if we follow Gofer's preferred sources, the state space {Heads,Tails}. But since it is almost never necessary to be as pedantic as Gofer wants us to be, many people equate the two.

AGAIN: The sample space is the set of all possible outcomes. The model of the sample space that we use, properly called the state space if you want to be that pedantic, is the set of all possible combinations for the set of random variables we choose to use to describe it.

Isn't it interesting that Gofer never tries to rebut this? That he only asks me to explain ambiguous references that he cherry-picks to maybe, possibly, if you ignore enough definitions, justifies his position? Which he never clearly states?

- JeffJo
- Intellectual
**Posts:**2526**Joined:**Tue Mar 10, 2009 11:01 am

### Re: Game Show Problem

JeffJo wrote:

> And yet still not recognizing the words "described by" or "can be."

Right, our coin toss in the real world is "described" by our model. And there "can be" other sample spaces in our model describing a coin toss in the real world. Surely this cannot be that hard to comprehend.

JeffJo wrote:

> if we follow Gofer's preferred sources, the state space {Heads,Tails}.

That's funny because the sentence I just quoted doesn't say "state space" but "SAMPLE SPACE".

JeffJo wrote:

> AGAIN: The sample space is the set of all possible outcomes.

.. in our model.

JeffJo wrote:

> Isn't it interesting that Gofer never tries to rebut this?

There's nothing to rebut because no source supports your position.

JeffJo wrote:

> That he only asks me to explain ambiguous references that he cherry-picks to maybe,

> possibly, if you ignore enough definitions, justifies his position? Which he never

> clearly states?

I have stated it "clearly" many times. Here it is again: If we wish to model an experiment in the real world using measure-theoretic mathematics, there are two objects that need to be properly defined: the sample space and the probability space. Random variables are not really needed though.

> And yet still not recognizing the words "described by" or "can be."

Right, our coin toss in the real world is "described" by our model. And there "can be" other sample spaces in our model describing a coin toss in the real world. Surely this cannot be that hard to comprehend.

JeffJo wrote:

> if we follow Gofer's preferred sources, the state space {Heads,Tails}.

That's funny because the sentence I just quoted doesn't say "state space" but "SAMPLE SPACE".

JeffJo wrote:

> AGAIN: The sample space is the set of all possible outcomes.

.. in our model.

JeffJo wrote:

> Isn't it interesting that Gofer never tries to rebut this?

There's nothing to rebut because no source supports your position.

JeffJo wrote:

> That he only asks me to explain ambiguous references that he cherry-picks to maybe,

> possibly, if you ignore enough definitions, justifies his position? Which he never

> clearly states?

I have stated it "clearly" many times. Here it is again: If we wish to model an experiment in the real world using measure-theoretic mathematics, there are two objects that need to be properly defined: the sample space and the probability space. Random variables are not really needed though.

- Gofer
- Intellectual
**Posts:**199**Joined:**Mon May 09, 2016 8:24 am

### Re: Game Show Problem

> Right, our coin toss in the real world is "described" by our model. And there "can be"

> other sample spaces in our model describing a coin toss in the real world. Surely

> this cannot be that hard to comprehend.

Not at all. I've been saying that for months. It is you who objected when I said that these "others" can comprise a "plethora."

> That's funny because the sentence I just quoted doesn't say "state space" but "SAMPLE SPACE".

But your source does. Random variables map the sample space to the state space. The difference is that a true sample space distinguishes every possible outcome from every other. And like I said before, and you ignored, the description that we conveniently call the sample space is really a state space, but it is too pedantic to use two names so we call it the sample space.

>JeffJo wrote:

>> AGAIN: The sample space is the set of all possible outcomes.

>... in our model.

Our model doesn't include all possibilities, just those we are interested in.

> If we wish to model an experiment in the real world using measure-theoretic mathematics,

One thing you never make clear is why we should wish to use measure-theoretic mathematics.

>... there are two objects that need to be properly defined: the sample space and the

> probability space.

Another thing you never make clear is what you think "well-defined" means here. The sample space, as you use the term, or state-space, as you ignore in your source, is "well-defined" by identifying mappings from the set of truly-all possibilities, to a set that includes only the aspects we are interested in. These are called random variables.

+++++

Just to try to make Gofer understand, if he will let himself:

All of the possible outcomes of an experiment form distinct elements of a sample space. Theoretically, each instance of the experiment produces an outcome that is distinct from every other one. (Wikipedia: “Each possible outcome of a particular experiment is unique, and different outcomes are mutually exclusive.”)

We can’t deal with such a set, so we instead describe each outcome by a collection of what, in simple probability theory, are called random variables. These can be, but don’t have to be, numbers. So, for example, a coin flip almost always is described by the values “Heads” and “Tails.” But if you need to use measure-theoretic tools, they do need to be numbers. There is no such need in the Game Show Problem.

Usually, there is no need to make the extreme terminology distinctions that Gofer seems to insist on. So different explanations may simplify some of these details. Some do it differently. I’ll just quote one article from Wikipedia, that supports my terminology:

> In probability theory and statistics, a probability distribution is a mathematical function

> that, stated in simple terms, can be thought of as providing the probability of occurrence

> of different possible outcomes in an experiment. For instance, if the random variable X is

> used to denote the outcome of a coin toss ('the experiment'), then the probability

> distribution of X would take the value 0.5 for X=Heads, and 0.5 for X=Tails.

See? Heads and Tails are values of the random variable X.

> A probability distribution is defined in terms of an underlying sample space, which is

> the set of all possible outcomes of the random phenomenon being observed. The

> sample space may be the set of real numbers or a higher-dimensional vector space,

> or it may be a list of non-numerical values; for example, the sample space of a coin

> flip would be {Heads ,Tails}.

See? The possible values of the random variable is what defines the sample space.

> A univariate distribution gives the probabilities of a single random variable taking on

> various alternative values; a multivariate distribution (a joint probability distribution)

> gives the probabilities of a random vector—a list of two or more random variables—

> taking on various combinations of values.

See? The values of the random variables are the arguments of the distribution function.

> other sample spaces in our model describing a coin toss in the real world. Surely

> this cannot be that hard to comprehend.

Not at all. I've been saying that for months. It is you who objected when I said that these "others" can comprise a "plethora."

> That's funny because the sentence I just quoted doesn't say "state space" but "SAMPLE SPACE".

But your source does. Random variables map the sample space to the state space. The difference is that a true sample space distinguishes every possible outcome from every other. And like I said before, and you ignored, the description that we conveniently call the sample space is really a state space, but it is too pedantic to use two names so we call it the sample space.

>JeffJo wrote:

>> AGAIN: The sample space is the set of all possible outcomes.

>... in our model.

Our model doesn't include all possibilities, just those we are interested in.

> If we wish to model an experiment in the real world using measure-theoretic mathematics,

One thing you never make clear is why we should wish to use measure-theoretic mathematics.

>... there are two objects that need to be properly defined: the sample space and the

> probability space.

Another thing you never make clear is what you think "well-defined" means here. The sample space, as you use the term, or state-space, as you ignore in your source, is "well-defined" by identifying mappings from the set of truly-all possibilities, to a set that includes only the aspects we are interested in. These are called random variables.

+++++

Just to try to make Gofer understand, if he will let himself:

All of the possible outcomes of an experiment form distinct elements of a sample space. Theoretically, each instance of the experiment produces an outcome that is distinct from every other one. (Wikipedia: “Each possible outcome of a particular experiment is unique, and different outcomes are mutually exclusive.”)

We can’t deal with such a set, so we instead describe each outcome by a collection of what, in simple probability theory, are called random variables. These can be, but don’t have to be, numbers. So, for example, a coin flip almost always is described by the values “Heads” and “Tails.” But if you need to use measure-theoretic tools, they do need to be numbers. There is no such need in the Game Show Problem.

Usually, there is no need to make the extreme terminology distinctions that Gofer seems to insist on. So different explanations may simplify some of these details. Some do it differently. I’ll just quote one article from Wikipedia, that supports my terminology:

> In probability theory and statistics, a probability distribution is a mathematical function

> that, stated in simple terms, can be thought of as providing the probability of occurrence

> of different possible outcomes in an experiment. For instance, if the random variable X is

> used to denote the outcome of a coin toss ('the experiment'), then the probability

> distribution of X would take the value 0.5 for X=Heads, and 0.5 for X=Tails.

See? Heads and Tails are values of the random variable X.

> A probability distribution is defined in terms of an underlying sample space, which is

> the set of all possible outcomes of the random phenomenon being observed. The

> sample space may be the set of real numbers or a higher-dimensional vector space,

> or it may be a list of non-numerical values; for example, the sample space of a coin

> flip would be {Heads ,Tails}.

See? The possible values of the random variable is what defines the sample space.

> A univariate distribution gives the probabilities of a single random variable taking on

> various alternative values; a multivariate distribution (a joint probability distribution)

> gives the probabilities of a random vector—a list of two or more random variables—

> taking on various combinations of values.

See? The values of the random variables are the arguments of the distribution function.

- JeffJo
- Intellectual
**Posts:**2526**Joined:**Tue Mar 10, 2009 11:01 am

### Re: Game Show Problem

JeffJo wrote:

> But your source does. Random variables map the sample space to the state space. The

> difference is that a true sample space distinguishes every possible outcome from

> every other. And like I said before, and you ignored, the description that we conveniently

> call the sample space is really a state space, but it is too pedantic to use two

> names so we call it the sample space.

That's even more funny because nowhere in the W. article on "random variables" occurs the phrase "state space".

JeffJo wrote:

> Our model doesn't include all possibilities, just those we are interested in.

which is why there "could be" other sample space in our model describing the coin toss.

JeffJo wrote:

> One thing you never make clear is why we should wish to use measure-theoretic mathematics.

Irrelevant! But you would do well excogitating the reasons for the development of measure-theoretic probabiltiy theory.

JeffJo wrote:

> Another thing you never make clear is what you think "well-defined" means here.

The same way the set {1,2,3} is well-defined!

JeffJo wrote:

> The sample space, as you use the term, or state-space, as you ignore in your source,

> is "well-defined" by identifying mappings from the set of truly-all possibilities,

> to a set that includes only the aspects we are interested in.

No, a random variable is a mapping, requiring both source and target to be well-defined.

> But if you need to use measure-theoretic tools, they do need to be numbers.

False! Look up the definition of "probability space" on W.

JeffJo wrote:

> For instance, if the random variable X is

> > used to denote the outcome of a coin toss ('the experiment'), then the probability

> > distribution of X would take the value 0.5 for X=Heads, and 0.5 for X=Tails.

>

> See? Heads and Tails are values of the random variable X.

See the words "used to denote"? It simply means that a random variable COULD be used to describe the sample space.

JeffJo wrote:

> See? The values of the random variables are the arguments of the distribution function.

Yeah, so? I've never denied that one could describe a probability space "backwards" by defining random variables and their distributions on it.

> But your source does. Random variables map the sample space to the state space. The

> difference is that a true sample space distinguishes every possible outcome from

> every other. And like I said before, and you ignored, the description that we conveniently

> call the sample space is really a state space, but it is too pedantic to use two

> names so we call it the sample space.

That's even more funny because nowhere in the W. article on "random variables" occurs the phrase "state space".

JeffJo wrote:

> Our model doesn't include all possibilities, just those we are interested in.

which is why there "could be" other sample space in our model describing the coin toss.

JeffJo wrote:

> One thing you never make clear is why we should wish to use measure-theoretic mathematics.

Irrelevant! But you would do well excogitating the reasons for the development of measure-theoretic probabiltiy theory.

JeffJo wrote:

> Another thing you never make clear is what you think "well-defined" means here.

The same way the set {1,2,3} is well-defined!

JeffJo wrote:

> The sample space, as you use the term, or state-space, as you ignore in your source,

> is "well-defined" by identifying mappings from the set of truly-all possibilities,

> to a set that includes only the aspects we are interested in.

No, a random variable is a mapping, requiring both source and target to be well-defined.

> But if you need to use measure-theoretic tools, they do need to be numbers.

False! Look up the definition of "probability space" on W.

JeffJo wrote:

> For instance, if the random variable X is

> > used to denote the outcome of a coin toss ('the experiment'), then the probability

> > distribution of X would take the value 0.5 for X=Heads, and 0.5 for X=Tails.

>

> See? Heads and Tails are values of the random variable X.

See the words "used to denote"? It simply means that a random variable COULD be used to describe the sample space.

JeffJo wrote:

> See? The values of the random variables are the arguments of the distribution function.

Yeah, so? I've never denied that one could describe a probability space "backwards" by defining random variables and their distributions on it.

- Gofer
- Intellectual
**Posts:**199**Joined:**Mon May 09, 2016 8:24 am

### Re: Game Show Problem

> That's even more funny because nowhere in the W. article on "random variables" occurs

> the phrase "state space".

Wikipedia, under the "Random variables" section of the article on "Probability space," says:

] A random variable X is a measurable function X: Ω → S from the sample space Ω

] to another measurable space S called the state space.

Many of these terms are interchangeable, or have narrow definitions when narrow fields-of-view are required (which is not the case in the GSP). The point is that you don't get to cherry-pick the ones that don't mention my usage, BUT DON'T CONTRADICT IT EITHER, to claim my usage is wrong. I have given ample evidence that my terminology is used, and you have not found anything that says it isn't. Just examples where it isn't mentioned.

>> One thing you never make clear is why we should wish to use measure-theoretic mathematics.

> Irrelevant!

If you insist we must use it, you do indeed need to justify why. But you don't even know what narrow field of view it uses, do you?

>> Another thing you never make clear is what you think "well-defined" means here.

>

> The same way the set {1,2,3} is well-defined!

And since you don't say what those number represents, you have not "defined" anything, let alone "well-defined" a sample space.

Hint: do provide such a definition, you need to define those values as the range of a random variable.

> A random variable is a mapping, requiring both source and target to be well-defined.

Which you don't do. And, according to Wikipedia, it is a mapping from the sample space to the state space.

>> But if you need to use measure-theoretic tools, they do need to be numbers.

>

> False! Look up the definition of "probability space" on W.

True! look up the definition of "measure" on W. And your favorite quote, from the "random variable" article, under the heading "Measure-theoretic definition":

] The most formal, axiomatic definition of a random variable involves measure theory.

] Continuous random variables are defined in terms of SETS OF NUMBERS...

> See the words "used to denote"? It simply means that a random variable COULD be used

> to describe the sample space.

And it also means that when the sample space is denoted {Heads, Tails}, that those are values of a random variable.

> the phrase "state space".

Wikipedia, under the "Random variables" section of the article on "Probability space," says:

] A random variable X is a measurable function X: Ω → S from the sample space Ω

] to another measurable space S called the state space.

Many of these terms are interchangeable, or have narrow definitions when narrow fields-of-view are required (which is not the case in the GSP). The point is that you don't get to cherry-pick the ones that don't mention my usage, BUT DON'T CONTRADICT IT EITHER, to claim my usage is wrong. I have given ample evidence that my terminology is used, and you have not found anything that says it isn't. Just examples where it isn't mentioned.

>> One thing you never make clear is why we should wish to use measure-theoretic mathematics.

> Irrelevant!

If you insist we must use it, you do indeed need to justify why. But you don't even know what narrow field of view it uses, do you?

>> Another thing you never make clear is what you think "well-defined" means here.

>

> The same way the set {1,2,3} is well-defined!

And since you don't say what those number represents, you have not "defined" anything, let alone "well-defined" a sample space.

Hint: do provide such a definition, you need to define those values as the range of a random variable.

> A random variable is a mapping, requiring both source and target to be well-defined.

Which you don't do. And, according to Wikipedia, it is a mapping from the sample space to the state space.

>> But if you need to use measure-theoretic tools, they do need to be numbers.

>

> False! Look up the definition of "probability space" on W.

True! look up the definition of "measure" on W. And your favorite quote, from the "random variable" article, under the heading "Measure-theoretic definition":

] The most formal, axiomatic definition of a random variable involves measure theory.

] Continuous random variables are defined in terms of SETS OF NUMBERS...

> See the words "used to denote"? It simply means that a random variable COULD be used

> to describe the sample space.

And it also means that when the sample space is denoted {Heads, Tails}, that those are values of a random variable.

- JeffJo
- Intellectual
**Posts:**2526**Joined:**Tue Mar 10, 2009 11:01 am

### Re: Game Show Problem

> Many of these terms are interchangeable, or have narrow definitions when narrow fields-of-view

> are required (which is not the case in the GSP).

I disagree! If we wish to analyze the problem measure-theoretically, then all those things need proper definitions.

> The point is that you don't get

> to cherry-pick the ones that don't mention my usage, BUT DON'T CONTRADICT IT EITHER,

I pick the coin-toss example supporting of my position, not yours. You need to provide proof where your theory is substantiated.

> to claim my usage is wrong. I have given ample evidence that my terminology is used,

only in the area of statistics, according to some definition, and not in strict math (measure-theoretic).

> and you have not found anything that says it isn't. Just examples where it isn't

> mentioned.

>

> >> One thing you never make clear is why we should wish to use measure-theoretic

> mathematics.

> > Irrelevant!

>

> If you insist we must use it, you do indeed need to justify why. But you don't even

> know what narrow field of view it uses, do you?

I don't insist we use it; but if we do, things need to be properly defined.

> >> Another thing you never make clear is what you think "well-defined" means here.

>

> >

> > The same way the set {1,2,3} is well-defined!

>

> And since you don't say what those number represents, you have not "defined" anything,

> let alone "well-defined" a sample space.

I don't need to. They just need to be distinct, and could be sets.

> Hint: do provide such a definition, you need to define those values as the range

> of a random variable.

>

> > A random variable is a mapping, requiring both source and target to be well-defined.

>

> Which you don't do.

Sure I do. I have demonstrated it many times.

> >> But if you need to use measure-theoretic tools, they do need to be numbers.

> >

> > False! Look up the definition of "probability space" on W.

>

> True! look up the definition of "measure" on W. And your favorite quote, from the

> "random variable" article, under the heading "Measure-theoretic definition":

>

> ] The most formal, axiomatic definition of a random variable involves measure theory.

> ] Continuous random variables are defined in terms of SETS OF NUMBERS...

That's because it says "CONTINUOUS random variables", my emphasis!

Besides, that section onward should really be part of the article on probability spaces.

In my opinion, W. gives a rather poor description of a measure-theoretic random variable, so here I provide a better one:

A function f:A→B is said to be a random variable if for every b∈BB there exists an a∈AA such that f(a)=b where AA and BB are sigma-algebras defined on A and B. Note that, since a is a set, f(a) needs to be defined, which however is trivial.

So a random variables is really nothing more than a measurable function.

> > See the words "used to denote"? It simply means that a random variable COULD be

> used

> > to describe the sample space.

>

> And it also means that when the sample space is denoted {Heads, Tails}, that those

> are values of a random variable.

No, they could be the values of a random variable.

> are required (which is not the case in the GSP).

I disagree! If we wish to analyze the problem measure-theoretically, then all those things need proper definitions.

> The point is that you don't get

> to cherry-pick the ones that don't mention my usage, BUT DON'T CONTRADICT IT EITHER,

I pick the coin-toss example supporting of my position, not yours. You need to provide proof where your theory is substantiated.

> to claim my usage is wrong. I have given ample evidence that my terminology is used,

only in the area of statistics, according to some definition, and not in strict math (measure-theoretic).

> and you have not found anything that says it isn't. Just examples where it isn't

> mentioned.

>

> >> One thing you never make clear is why we should wish to use measure-theoretic

> mathematics.

> > Irrelevant!

>

> If you insist we must use it, you do indeed need to justify why. But you don't even

> know what narrow field of view it uses, do you?

I don't insist we use it; but if we do, things need to be properly defined.

> >> Another thing you never make clear is what you think "well-defined" means here.

>

> >

> > The same way the set {1,2,3} is well-defined!

>

> And since you don't say what those number represents, you have not "defined" anything,

> let alone "well-defined" a sample space.

I don't need to. They just need to be distinct, and could be sets.

> Hint: do provide such a definition, you need to define those values as the range

> of a random variable.

>

> > A random variable is a mapping, requiring both source and target to be well-defined.

>

> Which you don't do.

Sure I do. I have demonstrated it many times.

> >> But if you need to use measure-theoretic tools, they do need to be numbers.

> >

> > False! Look up the definition of "probability space" on W.

>

> True! look up the definition of "measure" on W. And your favorite quote, from the

> "random variable" article, under the heading "Measure-theoretic definition":

>

> ] The most formal, axiomatic definition of a random variable involves measure theory.

> ] Continuous random variables are defined in terms of SETS OF NUMBERS...

That's because it says "CONTINUOUS random variables", my emphasis!

Besides, that section onward should really be part of the article on probability spaces.

In my opinion, W. gives a rather poor description of a measure-theoretic random variable, so here I provide a better one:

A function f:A→B is said to be a random variable if for every b∈BB there exists an a∈AA such that f(a)=b where AA and BB are sigma-algebras defined on A and B. Note that, since a is a set, f(a) needs to be defined, which however is trivial.

So a random variables is really nothing more than a measurable function.

> > See the words "used to denote"? It simply means that a random variable COULD be

> used

> > to describe the sample space.

>

> And it also means that when the sample space is denoted {Heads, Tails}, that those

> are values of a random variable.

No, they could be the values of a random variable.

- Gofer
- Intellectual
**Posts:**199**Joined:**Mon May 09, 2016 8:24 am

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