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