I seem to have difficulties into marrying the intuitive definition of a random variable to the measure theoretical definition.

For example, let's consider the classical case of measuring a person height. Intuitively, it is a (sort of Gaussian) random variable $X$ which takes values in an interval according to an integral of a density function. All is fine so far.

The measure theoretic definition says that $X$ is a measurable function $X:\Omega\rightarrow\mathbb R$, with $\Omega$ being called the "sample space of all possible outcomes", $(\Omega, \mathcal F, \mathbb P)$ the probability space and $(\mathbb R, \mathcal B(\mathbb R), \lambda)$ the set of real numbers with the corresponding Borel set and Lebesgue measure.

For a coin, a sample space like $\Omega=\{H,T\}$ makes sense intuitively.

I know that starting from the cdf $F$ of $X$, I can artificially construct a probability space on the unit interval, which if I remember well is $(\Omega, \mathcal F, \mathbb P) = ((0,1), \mathcal B((0,1)), \lambda\circ F^{-1})$, but what does $(0,1)$ have to do with the "sample space of all possible outcomes"?

So without further ado, my stupid question is:

What would be a natural probability space $(\Omega, \mathcal F, \mathbb P)$, as well as the definition of $X$ for my height measuring example above?

  • $\begingroup$ What is $\lambda$ in your question text? $\endgroup$ – Yatharth Agarwal Mar 12 at 1:30

The height of a person must be a nonnegative real number, so we can take $\Omega$ to be $[0, \infty)$ (you can argue about what the actual lower bound should be, but there's no harm in including some values that are not actually possible). $\mathcal F = \mathcal B(\Omega)$, and $X$ is the identity function $x \mapsto x$. The probability measure $P$ corresponds to the distribution of $X$, whatever it happens to be: $P(A)$ is the probability that a person's height is in the set $A$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.