Significance of Y = X^2 + 1 as random variable instead of X

[s3a]

Homework Statement

Let X be a random variable with the following probability distribution
X 0 1 2 3 4
f(x) 1/16 1/4 3/8 1/4 1/16

If another random variable Y = X^2 + 1 is formed, find the mean E[Y].

2. Relevant equation
E[x] = Σ_x [ x f(x)]

The Attempt at a Solution

I know how to compute the answer, however I do not understand what computing the expected value with X^2 + 1 instead of X as the random variable means.

Could someone please explain that to me?

 

[Ray Vickson]

Homework Statement

Let X be a random variable with the following probability distribution
X 0 1 2 3 4
f(x) 1/16 1/4 3/8 1/4 1/16

If another random variable Y = X^2 + 1 is formed, find the mean E[Y].

2. Relevant equation
E[x] = Σ_x [ x f(x)]

The Attempt at a Solution

I know how to compute the answer, however I do not understand what computing the expected value with X^2 + 1 instead of X as the random variable means.

Could someone please explain that to me?

You have a new random variable, Y, and are asked to compute its expected value. It just so happens that Y can be expressed in terms of X, but that is not the main issue: Y is a random variable all on its own. It has a distribution, a mean, a variance, etc.----anything that a random variable can have!

 

[s3a]

That makes sense, but then I need to ask: So, is the significance of the equation Y = X^2 + 1 just to describe how the random variable Y of some event B depends on the random variable X of some event A?

Or are both of the aforementioned random variables of the same event?

Basically, when solving a "word problem" (unlike the one I was doing - which just directly gave me the other random variable and told me to compute its expected value), what is accomplished by the introduction of a new random variable that is dependent on the previous one?

 

[LCKurtz]

Suppose you roll a pair of dice, a red one and a green one. Let $R$ be the outcome for the red toss and $G$ be the outcome for the green toss. Both $R$ and $G$ are uniform discrete random variables with sample space $\{1,2,3,4,5,6\}$. But perhaps you are analyzing craps and your real interest is in $R+G$. It would make sense to know its distribution function.

 

[Ray Vickson]

That makes sense, but then I need to ask: So, is the significance of the equation Y = X^2 + 1 just to describe how the random variable Y of some event B depends on the random variable X of some event A?

Or are both of the aforementioned random variables of the same event?

Basically, when solving a "word problem" (unlike the one I was doing - which just directly gave me the other random variable and told me to compute its expected value), what is accomplished by the introduction of a new random variable that is dependent on the previous one?

Perhaps you are operating a factory, and the demand for your product this month is some random variable, $X$. Quantities like production cost, overtime hours, material consumption, etc., are all dependent on demand and so are random variables that are functions of $X$. Or, you may be firing a cannon, and the angle of inclination is a random variable $\Theta$. Then the range of the projectile, $X$, will also be random, and will be a function of $\Theta$.

Wherever you can have functions, you can have random variables that are functions of other random variables.

 

[s3a]

Alright, and with regard to the terminology, but without over-complicating things, is a random variable a function which maps the elements from a certain sample space into any other set (that is not necessarily a subspace of the aforementioned sample space) whose value at any given moment is subject to variations?

 

[LCKurtz]

Alright, and with regard to the terminology, but without over-complicating things, is a random variable a function which maps the elements from a certain sample space into any other set (that is not necessarily a subspace of the aforementioned sample space) whose value at any given moment is subject to variations?

The "any other set" is the real numbers.

 

[s3a]

LCKurtz:
Is it always the real numbers? According to this video ( https://www.khanacademy.org/math/pr...random_variables_prob_dist/v/random-variables ), it seems that we can define any event (is "event" - as in the subspace of a sample space - the correct term?) by a real number (such as heads being 1 and tails being 0, for a coin toss), but this website ( http://www.math.uah.edu/stat/prob/Events.html ) says: "Often [so not always], a random variable takes values in a subset T of $ℝ^k$ for some $k∈ℕ^+$.

So, this tells me that not only does it not need to be the case that k = 1, as it seems you implied, but the output of a random variable for any input could be something other than a subset T of $ℝ^k$ for some $k∈ℕ^+$.

Could you please elaborate on this?

 

[LCKurtz]

Ray is the real expert on this. I have read references that say yes, it is always the reals, and others that have a more general setting. What I am very sure about is that for someone just learning the material, for all practical purposes the answer is yes, it is always the reals. You aren't going to encounter anything else any time soon, if ever.

 

[s3a]

Okay, it's good to know it will almost always be the case that I'll be dealing with a set T (to again use the notation of this link: http://www.math.uah.edu/stat/prob/Events.html) that is a subset of the real numbers, but to be very specific, it seems to me that, in general, the set T could be a set containing anything such as rabbits and triangles, for example (instead of real numbers), such that, in the case of the coin toss, I could have a random variable X being a mapping of heads to rabbit and tails to triangle (as opposed to heads to 1 and tails to 0).

Ray Vickson, is what I said above, in this post, correct?

P.S.
Sorry for being so specific.

P.P.S.
Ray Vickson, it makes sense that some would define a random variable as only outputting a real number since such an output could always be linked to whatever is being dealt with.

 

[s3a]

Also, to reiterate, is it correct to say that the random variable is mapping an event (as in a subspace of the sample space in question) to some set T?

 

[LCKurtz]

Look here, for example:
http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm

 

[Ray Vickson]

LCKurtz:
Is it always the real numbers? According to this video ( https://www.khanacademy.org/math/pr...random_variables_prob_dist/v/random-variables ), it seems that we can define any event (is "event" - as in the subspace of a sample space - the correct term?) by a real number (such as heads being 1 and tails being 0, for a coin toss), but this website ( http://www.math.uah.edu/stat/prob/Events.html ) says: "Often [so not always], a random variable takes values in a subset T of $ℝ^k$ for some $k∈ℕ^+$.

So, this tells me that not only does it not need to be the case that k = 1, as it seems you implied, but the output of a random variable for any input could be something other than a subset T of $ℝ^k$ for some $k∈ℕ^+$.

Could you please elaborate on this?

Different sources may use different nomenclature. Basically, though, a random variable $X$ is a variable (in some space) which occurs "at random", so is essentially a mapping from $\Omega$ (the "sample space") to an outcome (or value) space ${\cal X}$. That is, for every $\omega \in \Omega$ we have a value (or observation, or outcome) $X(\omega) \in {\cal X}$. If ${\cal X} = \mathbb{N}$ we have an integer-valued random variable. If ${\cal X} = \mathbb{R}$ we have a real-valued random variable. If ${\cal X} = \mathbb{R}^n$ we have a n-dimensional random variable = a random vector.

Mostly, we do not deal with an explicit sample space $\Omega$, but, instead, with constructs like density/distribution functions. However, in principle, if we have a function $y = h(x)$ the random variables $X, Y$ (connected through $Y = h(X)$) are on the same sample-space. Again, though, we often do not need to know, nor do we care about, what the sample space actually is.

 

[s3a]

So, to say it in my own words, is the following correct?:
A random variable is a function mapping a sample space to an outcome space?

Edit: I just realized that you wrote exactly what I wrote (with a few extra symbols) above. :P

 

[Ray Vickson]

So, to say it in my own words, is the following correct?:
A random variable is a function mapping a sample space to an outcome space?

Edit: I just realized that you wrote exactly what I wrote (with a few extra symbols) above. :P

I think, more explicitly, that a random variable is numerical-valued (in one dimension or many, real or discrete), because we typically associate concepts such as distribution or density functions, means, variances, etc. If you had random colors, for example, it would be hard to know what mean color, variance of color, cumulative distribution of color, etc, could possibly mean. So, I really don't know if one had an outcome space like ${\cal X} = \{\text{red, blue, cyan, green , yellow, purple, orange}\}$, whether or not we would regard a mapping $\Omega \longrightarrow {\cal X}$ as a random variable, or not. Certainly, colors can be selected at random, but I'm not convinced that would make them outcomes of a random variable. However, that is just my personal opinion/confujsion.

 

[s3a]

Okay, so in that case, do you think it would be correct to say that "A random variable is a function mapping the sample space in question to a real-number representation of a random outcome, where the outcome is an event."?

 

[LCKurtz]

I think, more explicitly, that a random variable is numerical-valued (in one dimension or many, real or discrete), because we typically associate concepts such as distribution or density functions, means, variances, etc. If you had random colors, for example, it would be hard to know what mean color, variance of color, cumulative distribution of color, etc, could possibly mean. So, I really don't know if one had an outcome space like ${\cal X} = \{\text{red, blue, cyan, green , yellow, purple, orange}\}$, whether or not we would regard a mapping $\Omega \longrightarrow {\cal X}$ as a random variable, or not. Certainly, colors can be selected at random, but I'm not convinced that would make them outcomes of a random variable. However, that is just my personal opinion/confujsion.

I guess that is an example where you might think as $\cal X$ as $\mathbb R^3$ where the color choice is expressed as the $(R,G,B)$ intensities (numbers in $[0,1]$) that define color. Can't say I've ever dealt with such random variables though.

 

[s3a]

I don't know if it's because I'm tired as I write this, but I'm getting very confused now.

Is the outcome space an event?

Having said that, the (R, G, B) intensities example seems like a nice example, but I still want to formulate an easy-to-understand and 100%-accurate definition for myself.

 

[LCKurtz]

I don't know if it's because I'm tired as I write this, but I'm getting very confused now.

Is the outcome space an event?

Having said that, the (R, G, B) intensities example seems like a nice example, but I still want to formulate an easy-to-understand and 100%-accurate definition for myself.

The sample space for an experiment is the set of all possible outcomes of that experiment. I guess that's what you mean when you say "outcome space". It's best not to make up your own terms because it adds to confusion. An event is a subset of the sample space. So if $S$ is a sample space and $E\subset S$ is an event, if the outcome of an experiment is in $E$, we say the event $E$ occurred.

 

[s3a]

Actually, I got the term "outcome space" from Ray Vickson in this thread, and post #2 here ( https://www.physicsforums.com/threads/outcome-space-and-sample-space.434220/ ) explains what it is, but I think we're overcomplicating things.

Before I say anything else, is it possible to compute the expected value of (R, G, B) = (0.1, 0.2, 0.3), for example (since we're using $ℝ^3$ instead of colour names)?

 

[s3a]

Alright, I think I'm going to settle with a simple definition that I found in a book I just looked at which says the following.:
"A random variable is a function from the sample space S into the set of real numbers R."

Sorry for the pedantic nature of this thread, and thank you both for your input.

 

[LCKurtz]

Alright, I think I'm going to settle with a simple definition that I found in a book I just looked at which says the following.:
"A random variable is a function from the sample space S into the set of real numbers R."

Like I said in posts 7 and 9...

 

[s3a]

Yes, but I took what you had said as your experience of what tends to happen as opposed to it being a solid definition given in a textbook.