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 $X.$

  1. Proposition: Monotonically Increasing Property of Probability Distributions

Proofs: 1 2 3
Propositions: 4 5 6

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  1. Hedderich, J.;Sachs, L.: "Angewandte Statistik", Springer Gabler, 2012, Vol .14