What does it mean a random variable to follow a certain distribution?

Question

Let $X$ be a random variable. For example, what does it mean $X$ to follow the binomial distribution $\operatorname{binom}(n,p)$ ($X\sim\operatorname{binom}(n,p)$)?

Answer 1

In my opinion a hidden but crucial fact (or rather definition) behind this is the notion of random variable and its extended meaning in combination with a probability space. Technically, a random variable $X$ is a measurable function $\Omega \to E$ from a measurable space $(\Omega, \mathcal F)$ to another measurable space $(E,\mathcal E)$. Whenever the notion of distribution of a random variable comes into play, one also assumes that there is a probability measure $\Bbb P$ defined on $(\Omega , \mathcal F)$. Therefore, $(\Omega , \mathcal F, \Bbb P)$ is a so-called probability space.

We can now define the so-called pushforward measure of $\Bbb P$ by $X$ (denoted by $\Bbb P\circ X^{-1}$): For $A\in \mathcal E$ we set $$\Bbb P \circ X^{-1} (A) := \Bbb P ( X^{-1} (A) ),$$ which is in this case again a probability measure. This works since $X: (\Omega, \mathcal F) \to (E,\mathcal E)$ is measurable. In the same way as measurable function was relabelled to random variable, in probability theory we speak of $\Bbb P\circ X^{-1}$ as the distribution of $X$.

If $\mu$ is a probability measure (for example Binomial distribution), then to say that $X$ follows this distribution (denoted by $X\sim \mu$) means that $$\Bbb P\circ X^{-1} = \mu.$$

If $\mu$ is the Binomial distribution with parameters $n$ and $p$, this means that for $k= 0, \ldots, n$ we have $$\binom n k p^k (1-p)^{n-k} = \Bbb P\circ X^{-1} (\{k\}) = \mathbb P (X = k)$$