Geometric distribution
High school
Consider a sequence of independent trials, each of which succeeds with probability $p$. The number $N$ of trials up to and including the first success has a geometric distribution with parameter $p$:
\[P(N = k) = (1 - p)^{k - 1} p \quad \text{for } k = 1, 2, 3, \dots.\]Properties
- A geometric random variable with parameter $p$ has mean $1 / p$.
- Some authors instead count the failures before the first success, which takes values $0, 1, 2, \dots$ and has mean $(1 - p) / p$.
- The geometric distribution is memoryless: $P(N > j + k \mid N > j) = P(N > k)$.
- The geometric distribution is the discrete counterpart of the exponential distribution.