probability distribution

the probability distribution is a function that gives the probabilities of different outcomes for an experiment.
if $\text{[math]}$ is a discrete random variable then the list of values that it could receive and their corresponding probabilities are called the distribution law, probability law, or the distribution of said variable

let $\text{[math]}$ be a random variable.

the probability measure $\text{[math]}$ is called the distribution of $\text{[math]}$ .
for a real random variable $\text{[math]}$ , the map $\text{[math]}$ is called the distribution function of $\text{[math]}$ (or, more accurately, of $\text{[math]}$ ). we write $\text{[math]}$ if $\text{[math]}$ and say that $\text{[math]}$ has distribution $\text{[math]}$ .
a family $\text{[math]}$ of random variables is called identically distributed if $\text{[math]}$ for all $\text{[math]}$ . we write $\text{[math]}$ if $\text{[math]}$ ( $\text{[math]}$ for distribution).

[cite:;taken from @klenke_prob_2020 definition 1.103]

[cite:;see @klenke_prob_2020 definition 1.59]