Geometric distribution

Definition

Let $p \in (0, 1]$. The geometric distribution with parameter $p$ is the distribution on $\{0, 1, 2, \dots\}$ with the probability mass function $$p(k) = (1 - p)^k \, p \quad \text{for } k = 0, 1, 2, \dots$$

Remarks

It is the distribution of the number of failures before the first success in independent Bernoulli trials with success probability $p$. The other common convention counts the trials up to and including the first success. That number $Y = X + 1$ has $P(Y = k) = (1 - p)^{k-1} p$ for $k \ge 1$. For $X$ as in the definition, $P(X \ge k) = (1 - p)^k$, the mean is $(1 - p)/p$ and the variance is $(1 - p)/p^2$. The geometric distribution is the only distribution on $\{0, 1, 2, \dots\}$ with the memoryless property.

Explanations

L1 human

Toss a coin until the first head. The number of tails before it has the geometric distribution. Each extra tail multiplies the probability by $1 - p$, so the probabilities form a geometric sequence.