Geometric Distribution

Definition

$X\sim \text{Geom}(p)$

Special case of negative binomial distribution with $r=1$. Distribution for number of failures needed for a single success.

• pmf: $(1-p{)}^{x}p$, $x\in 0,1,2,\dots$
• mean: $\frac{1-p}{p}$
• var: $\frac{1-p}{p^2}$
• mgf: $\frac{p}{1-(1-p)e^t}$, $t<-\ln (1-p)$