Factsheet: Geometric distribution
Where to use: The geometric distribution is used to count \(X\), the number of Bernoulli trials until a successful outcome is reached.
Notation: \(X \sim \textrm{Geometric}(p)\)
Parameter: \(p\) is the real number representing the probability of success in a single trial (where \(0 \le p \le 1\)).
| Quantity | Value | Notes |
|---|---|---|
| Mean | \(\mathbb{E}(X) = \dfrac{1}{p}\) | |
| Variance | \(\mathbb{V}(X) = \dfrac{1-p}{p^2}\) | |
| PMF | \(\mathbb{P}(X=x)=(1-p)^{k-1}p\) | |
| CDF | \(\displaystyle\mathbb{P}(X\leq x)=\begin{cases} 1-(1-p)^x & \textsf{if } x\geq0 \\0 & \textsf{if } x<0\end{cases}\) |
Example: You flip a coin multiple times, and the probability of getting ‘heads’ is \(0.5\). You decide to stop flipping the coin once you get a ‘heads’. Taking ‘heads’ as a success, this can be expressed as \(X \sim \textrm{Geometric}(0.5)\). It means the probability of success is \(0.5\), and you will stop conducting trials after you reach a success.
Further reading
Version history
v1.0: initial version created 04/25 by tdhc and Michelle Arnetta as part of a University of St Andrews VIP project.
v1.1: moved to factsheet form and populated with material from Overview: Probability distributions by tdhc.
v1.2: deprecated shinylive apps replaced 09/26 by tdhc.