# Geometric Distribution

We use the Geometric Distribution to answer the following question:

- Given a probability of success $p$ per trial, what is the probability that our first success is on the $k$-th trial, i.e. $P(X=k)$?

Geometric Distribution

$X∼Geo(p)$ where $X$ is the number of trials required to observe the first success in a sequence of independent Bernoulli experiments.

This can also be formulated as “number of fails before 1st success”.

Suppose $X∼Geo(p)$,

- Support of $X$: ${1,2,3,…}$
- Probability Mass Function of $X$: $Pr(X=k)=(1−p)_{k−1}p$

The intuition is simple. You have a probability of failing with $(1−p)$ over $(k−1)$ trials, and a probability of success of $p$ of succeeding.

- $E(X)=p1 $
- $Var(X)=p_{2}1−p $