Let \(X\) be a random variable of a given random experiment. Assume, we have a random experiment, for which the probability of the event.
"$X$ has a realization less or equal a given real number \(x\)",
i.e. the probability $p(X \le x)$ exists for all real numbers $x\in\mathbb R.$ Then we call the function. $$f:=\cases{\mathbb R\mapsto[0,1]\\x\mapsto p(X\le x)}\quad\quad\text{for all }x\in\mathbb R$$
the probability distribution of the random variable $X.$
Proofs: 1 2 3
Propositions: 4 5 6