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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?

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  • $\begingroup$ What is $\lambda$ in your question text? $\endgroup$ Mar 12, 2019 at 1:30

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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 $A$.

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