# Proposition: Geometric Distribution

We repeat a Bernoulli experiment (if necessary, infinitely many times). Each time, we observe, if an event $$A$$ occurred or not. Let the probability of $$A$$ to occur be constantly $$p:=p(A)$$ during each repetition of the experiment. Let $$X$$ be the random variable having the realization $$k$$, if and only if we observe $$A$$ for the first time at $$k$$-th repetition of the experiment. Then, the probability mass function is given by

$p(X = k)=\begin{cases} p(1-p)^{k-1}&\text{for }k=1,2,3,\ldots \\\\ 0&\text{else.}\end{cases}$

The geometric distribution (i.e. the probability distribution of the random variable $$X$$) is given by

$\begin{array}{rcll} p(X \le x)&=&0&\text{for }x < 1\\ p(X \le x)&=&\sum_{k=1}^{k=x}p(1-p)^{k-1}&\text{for }1\le x \end{array}$

In the following interactive you can change the probability $$p$$ of observing an event $$A$$ in a Bernoulli experiment repeated up to $$30$$ times. Changing this probability will change the probability mass function for different values of $$X$$ (red) of observing $$A$$ exactly in the $$k$$-th repetition, and the probability distribution (blue):

Proofs: 1

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