Distribution

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

$X\sim Geo(p)$ 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”.

• Saint-Petersburg Paradox

Suppose $X\sim Geo(p)$,

1. Support of $X$: $\{1,2,3,\dots \}$
2. Probability Mass Function of $X$: $\mathrm{Pr}(X=k)=(1-p{)}^{k-1}p$

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.

• $E(X)=\frac{1}{p}$
• $Var(X)=\frac{1-p}{p^2}$

Concepts

• Memorylessness