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## Definition: Probability Distribution

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 $X.$

| | | | | created: 2016-03-19 19:14:42 | modified: 2020-04-17 09:07:20 | by: bookofproofs | references: [1796]