I'm struggling with some basic concepts regarding the definitions of random variables.

If I'm not mistaken, random variables are functions which aim to "translate" outcomes of a sample space to a measurable space. As an example, the sample space composing the possible outcomes of a coin toss would be "head" or "tails". By assigning a mathematical function to these values, we can define them as ${[0,1]}$, allowing further analysis such as inferring the probability distribution of outcomes along multiple trials and the expected value of such a distribution.

The necessity of defining discrete outcomes as a random variable is pretty clear to me. What I do not understand is what kind of transformation a interval within a continuous sample space undergoes to be defined as a continuous random variable. Is there any difference between these two definitions (continuous sample space x continuous random variable)? Why do we define the probability distribution of a continuous random variable, instead of the probability distribution of a continuous sample space? Is it just a formalization?

  • 3
    $\begingroup$ You seem to have misinterpreted some closely-connected concepts. Technically speaking, there are no "continuous/discrete sample space", but "uncountable/countable sample space" (just like you only say "continuous functions" but not "continuous sets"). Also, we only say "distribution of random variables", but never "distribution of sample spaces". $\endgroup$
    – Zhanxiong
    Nov 14, 2022 at 22:06

1 Answer 1


First, a basic commentary:

$\bullet$ Let $(\Omega, \boldsymbol{ \mathfrak A}, \mu)$ be a measure space. Let $f$ be a real-valued function on $D\in \boldsymbol{ \mathfrak A}.$ Consider another measure space $(\Omega^\prime, \boldsymbol{ \mathfrak B}).$ $f$ is $\boldsymbol{ \mathfrak A}$-measurable on $D$ if and only if $f$ is $\boldsymbol{ \mathfrak A}/\boldsymbol{\mathfrak B}$ measurable mapping of $D$ into $\Omega^\prime$ that is, $f^{-1}(\boldsymbol{\mathfrak B})\subset \boldsymbol{ \mathfrak A}.$

$\bullet$ Let $(\Omega^\prime, \boldsymbol{ \mathfrak A^\prime})$ be a measurable space; let $X:\Omega \to \Omega^\prime$ be measurable. $X$ is a random variable with $(\Omega^\prime, \boldsymbol{ \mathfrak A^\prime}) = (\mathbb R, \boldsymbol{ \mathfrak B}_\mathbb R).$

$\bullet$ Let $(\Omega, \boldsymbol{ \mathfrak A}, \mathbf P)$ be the probability space. Distribution of $X$ is the image measure of $\mathbf P$ under $X,$ that is, $\mathbf P_X:= \mathbf P\circ X^{-1}.$

$\bullet$ $X$ is continuous if $\mathbf P_X$ is absolutely continuous. For example, let $X$ be a real random variable. It is normal if $\mathbf P_X:= \mathcal N_{\mu,\sigma^2} = f\boldsymbol\lambda$ where $f(x) = (\sqrt{2\pi\sigma^2})^{-1}\exp\left[-{(x-\mu)^2}/{2\sigma^2}\right]$ and $\boldsymbol\lambda$ is Lebesgue measure on $\mathbb R.$

Now, we are not interested with the underlying probability space $(\Omega, \boldsymbol{ \mathfrak A}, \mathbf P):$

Events of $\Omega$ are not observed directly. Rather, the observations are aspects of the single experiments that are coded as measurable maps from $\Omega$ to a set of possible observations.

Probabilities of those observations are measured by the image measure $\mathbf P_X;$ it is of vital interest. It determines whether $X$ is of continuous type. What happens to $\omega \mapsto \omega^\prime$ is not the question we bother with.


$\rm [I]$ Probability Theory: A Comprehensive Course, Achim Klenke, Springer, $2020,$ sec. $1.5,$ pp. $45-50.$


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