Geometric Distribution#

Definition#

Definition 37 (Geometric Distribution)

Let \(X\) be a Geometric random variable. Then the probability mass function (PMF) of \(X\) is given by

\[ \P \lsq X = k \rsq = (1-p)^{k-1} p \qquad \text{for } k = 1, 2, \ldots \]

where \(0 \leq p \leq 1\) is called the geometric parameter.

We write

\[ X \sim \geometric(p) \]

to say that \(X\) is drawn from a geometric distribution with parameter \(p\).

Property 13 (Expectation of Geometric Distribution)

Let \(X \sim \geometric(p)\) be a Geometric random variable with parameter \(p\). Then the expectation of \(X\) is given by

\[ \expectation \lsq X \rsq = \sum_{k=1}^{\infty} k \cdot \P \lsq X = k \rsq = \frac{1}{p} \]

Property 14 (Variance of Geometric Distribution)

Let \(X \sim \geometric(p)\) be a Geometric random variable with parameter \(p\). Then the variance of \(X\) is given by

\[ \var \lsq X \rsq = \expectation \lsq X^2 \rsq - \expectation \lsq X \rsq^2 = \frac{1-p}{p^2} \]