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My understanding is that a random variable $X$ is a function that goes from a sample space $\Omega$ to a measurable space $E$, where $\Omega$ is part of a probability space (along with a sigma-algebra $F$ and a measurable function $\mu$) and where $E$ is tipically $\mathbb R$. A sample space is a set of all the possible "outcomes" that may arise in an experiment, $F$ could be thought of as a set of "events" (which are, in turn, sets of outcomes), and $\mu$ is a measure used to assign probabilities to events in $F$.

I am not a mathematician, and I only have very superficial knowledge of Probability Theory and Measure Theory. Nonetheless, I am interested on having a general intuition of the concepts I've mentioned above, and to learn how to correctly use the vocabulary of Probability and Statistics. Thus, I am concerned about the way the terms "Outcome" and "Sample Space" seem to be used in different places; particularly when it's implied that the codomain or the image of $X$ is the Sample Space of $X$, and/or that $X$ "takes the values of" the "outcomes" (which I believe are both wrong given the definitions above).

Some examples include:

  1. Wikipedia's article on "Random Variates":

    A random variate is a particular outcome of a random variable

Reading ahead it's clear what they mean: that a Random Variate is a particular value that a r.v. can take. Thus, that sentence says that an "outcome" is a possible value for a r.v., and not a value where the r.v., as a function, is evaluated.

  1. An answer at Cross Validated, on the nature of "Sample" and "Outcome":

    So the Sample Space will be "{Heads, Tails}", which will be the domain of the random variable, while the "outcome space", its range, will be {5,17}

Since the "outcome space" is the range of a r.v., then "outcomes" are elements of the range and not of the domain, according to his definition.

  1. Wikipedia's article on PDF's:

    In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function, whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as […]

I always thought intuitively that a PDF helps in solving problems like finding the probability that $ X \le k $, by making integrals from minus infinity to k. Thus, the domain of a PDF is the image of X, and not the domain of X, right?

  1. The one that confused me the most (although I can't find it now) was one that said something along the lines of "X is a discrete r.v. if its sample space is discrete". I believe that's totally wrong if we define the sample space of X as its domain!

So my conclusion is that it's extremely common to find the terms "sample space" and "outcome" used to refer to the image of a random variable and its possible values. Nonetheless, that usage is fundamentally different from the one used in more "formal" treatments of probability theory, and one must be always alert as to what definition is being used.

So my questions are:

  1. Is my conclusion right, or am I missing something?
  2. Is it possible to define a r.v. where both its domain and its image are the same sample space? I'm unsure if that even makes sense, but that may make it acceptable to say things like the ones I've cited. I'm specially interested in that if $\Omega$ and $E$ can both be $\mathbb R$?!
  3. Is it possible to make a probability space "around" $E$ (the codomain of X)? So that $E$ is also a Sample Space?

Thanks!

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The confussion is exactly that while a random variable maps a sample space onto a measurable space, that measurable space is also a sample space.   It has all the requisite properties of one.

So it is permissable to refer to the values of a random variable as outcomes.   It may cause some confusion, if you are not very careful, but it is permissable.

So

  1. Yes.   It is somewhat common to find the terms "sample space" and "outcome" used to refer to the image of a random variable and its possible values.

  2. Yes.   It is possible to define a r.v. where both its domain and its image are the same sample space.   One such random variable would be the identity function; though there are others.   Consider a sample space consisting of the results shown on a die roll, and the random variable of seven minus the results shown on the die roll. $X:x\mapsto 7-x$. $$X: \{1,2,3,4,5,6\}\mapsto\{1,2,3,4,5,6\}$$

  3. Yes.   The image of a random variable can be associated with a sigma algebra and a probability measure.   Consider the results of the toss of two fair coins, and the random variable that is the count of heads among them.   The random variable's image $\{0,1,2\}$ is the measurable space mapped onto by the random variable, we can build a sigma algebra (its powerset for example) and these events can be assigned a probability measure. $$\mathsf p(A)= \begin{cases}1 & :& A\supseteq \{1,2,3\}\\ 2/3&:& A\cap\{1,2,3\}\in \{\{1,2\}, \{1,3\}, \{2,3\} \}\\ 1/3 &:& A\cap\{1,2,3\}\in \{\{1\}, \{2\}, \{3\} \} \\ 0 &:& A\cap\{1,2,3\}=\emptyset\end{cases}$$ The triple, $(\{1,2,3\}, 2^{\{1,2,3\}}, \mathsf p)$ , fits the criteria required to be a probability space.

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Measure space gives weights to every set in an intuitively consistent way as expressed by the axioms of a measure. Probability measures give weight 1 to whole space.

A random variable is a function that has some bias towards some numbers. If your experiment is selecting a child randomly from a play school for kids aged less than 6, and if the random variable is the weight of the child, it has a bias to values less than 40 kg. (That is what the distribution function of the r.v. tries to capture). Here your sample space those kids in that school.

If another experiment chooses the first passenger that comes out of a commuter train at 9 am in a particular railway station, the sample space is different (adults will be there in good numbers) and the weight random variable here will take values above 50 kg much more often.

I have answered a slightly different but closely related question here and it should clarify many things: https://mathoverflow.net/questions/250500/why-do-we-need-random-variables/252491#252491

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