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In class, a teacher told me that the realization of a random variable is also a random variable. For example, if I take the a sample mean, and that mean results in the value 35, then 35 is also a random variable. In a book (pdf page 181), I also found something similar:

Throughout this chapter we consider a sequence of random variables X1, X2, X3, ... . You should think of Xi as the result of the ith repetition of a particular measurement or experiment

Here I interpret "thinking of Xi as the result of the ith repetition" as Xi being the realization of the random variable Xi, but in the previous sentence, the author is including Xi in the set of random variables.

Why would this be the case?

I find that the realization of a random variable also being a random variable is at odds with the definition of a random variable being a function that maps from the sample space of the outcomes of an experiment to the real values. When a RV is, for example, 35, there is no function.

For me it is intuitive to think of random variables and the realizations of random variables as separate things.

What am I getting wrong?

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    $\begingroup$ $\frac{33+37}{2}=35$ is a number that lacks randomness (arguably a degenerate random variable, sure). $\frac{X_1+X_2}{2}=\bar X$ does have randomness. I’d ask the instructor for clarification. “Professor, what is the distribution function for the 35?” $\endgroup$
    – Dave
    Commented Feb 24 at 17:14
  • $\begingroup$ Sure, but what about the book. Don't you interpret that it is saying something similar? $\endgroup$ Commented Feb 24 at 20:30
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    $\begingroup$ Your intuition and definitions sound good, but you seem to conflate "realization of" with "function of." The mean, qua function of the random variables $X_1,\ldots,X_n,$, is a random variable. The mean of the realization is the mean of a bunch of numbers and, as such, is just a number. $\endgroup$
    – whuber
    Commented Feb 24 at 21:50

2 Answers 2

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A function of a random variable (or of several random variables) is also a random variable and so has a distribution. Your book on page 42 introduces the idea that a random variable is a function $\Omega \to \mathbb R$, and clearly a further function $\mathbb R \to \mathbb R$ applied to this would give another function $\Omega \to \mathbb R$ which could then be taken to be another random variable.

So in your example, $\bar X_n = \dfrac{X_1+X_2+\cdots +X_n}{n}$ is a random variable, and here $35$ is a realization of $\bar X_n$.

In your book, this is implicit. On page 197, it is more explicit when it defines $Z_n = \sqrt{n}\dfrac{\bar X_n - \mu}{\sigma}$ and says $Z_n$

has expected value $0$ and variance $1$. What more can we say about the distribution of the random variables $Z_n$?

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Constants can be considered as random variables if we like

When dealing with probability theory and random variables, there are two things we might mean by a "constant". One thing we might mean by a "constant" is a value $x \in \mathbb{R}$ that is some real number.$^\dagger$ This is not a function from a sample space so it is not a random variable. Another thing we might mean by a "constant" is a function $x_*: \Omega \rightarrow \mathbb{R}$ that maps all the elements of the sample space to that same real number (i.e., with $x_*: \omega \mapsto x$ for all $\omega \in \Omega$ and some $x \in \mathbb{R}$). This is a random variable which has a point-mass distribution on the value $x \in \mathbb{R}$. (I have used the notation $x_*$ here to help to explain this function, but in practice we would probably use an abuse-of-notation and call it $x$, but also call the mapped real number $x$ as well, even though these things are mathematically different objects.)

Since it is possible to define a "constant" that is actually a random variable, when we do probabilistic analysis, we generally give ourselves the latitude to use any "constant" in our analysis as if it were a random variable with a point-mass distribution. Technically speaking, it turns out that it is coherent to do this --- you can consider an observed random variable that is now known to be equal to a particular value to be a random variable with a point-mass distribution on that value. The explanation for this is a bit technical, owing to the fact that conditional probability is formally defined through measure theory (which is a bit far removed from the material that most students of probability first learn). I will give an explanation here for completeness.

Suppose we have a random variable $X$ with some distribution function $F_X$. Now suppose that we observe the event $\{ X=x \}$ and we want to describe the conditional distribution of $X$ given observation of this event --- i.e., we want to find the probability measure $\mathbb{P}(X \in \mathcal{A} | X=x)$ for any event $\mathcal{A}$. If we try the candidate:

$$\mathbb{P}(X \in \mathcal{A} | X=x) = \mathbb{I}(x \in \mathcal{A}),$$

then for any event $\mathcal{A}$ we have:

$$\begin{align} \mathbb{P}(X \in \mathcal{A}) &= \int \limits_\mathcal{A} dF_X(x) \\[6pt] &= \int \limits_\mathbb{R} \mathbb{I}(x \in \mathcal{A}) dF_X(x) \\[6pt] &= \int \limits_\mathbb{R} \mathbb{P}(X \in \mathcal{A} | X=x) dF_X(x), \\[6pt] \end{align}$$

which satisfies the defining requirement for a conditional probability distribution. This means that the conditional probability measure we have formulated is a valid conditional distribution for $X$ given $\{X=x\}$. This conditional distribution is a point-mass on the value $x$ so it is equivalent in distribution to a "constant" random variable $X_*$ defined by the mapping $X_*: \omega \mapsto x$. This is the essence of what allows us to replace a non-degenerate random variable with a "constant" random variable once we observe it to be equal to a particular value. Because we can always do this, practitioners are usually pretty loose with this practice, including using a standard abuse-of-notation whereby we use the same notation for a real number and a random variable that always maps to that real number (the two types of "constants" discussed above).

As to the further issues of random "realisations" of a particular thing, I think what your teacher has in mind here is for $X_1,X_2,X_3,...$ to be a sequence of realisations of a particular repeatable thing (which one might describe by some generic $X$ that refers to the whole sequence). Saying that $X_i$ is a realisation of the random variable $X_i$ doesn't make much sense, but you could say that $X_i$ is the $i$th realisation of a particular repeatable thing described by this sequence of random variables. Again, if you were to observe any of these random variables then it would be coherent to now treat them as "constants" in the sense of being random variables with a point-mass distribution on their known values.


$^\dagger$ I will work here with real random variables but it is of course possible to have complex random variables or other types.

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