Geometric Distribution

Compact study note.

Summary

The geometric distribution models the trial number of the first success in independent Bernoulli trials.^[1]

$X \sim Geometric (p)$ here means $X$ counts trials until the first success.

Trials are independent and each has success probability $p$ .

$0 < p \leq 1$ .

${1, 2, 3, \dots}$ .

$P (X = k) = (1 - p)^{k - 1} p$ for $k = 1, 2, \dots$ .

$F_{X} (k) = 1 - (1 - p)^{⌊ k ⌋}$ for $k \geq 1$ .

Moments and MGF:

E [X] = 1 / p, Var (X) = (1 - p) / p^{2} .

M_{X} (t) = \frac{p \exp (t)}{1 - (1 - p) \exp (t)}, t < - \log (1 - p) .

The geometric distribution is memoryless: $P (X > m + n ∣ X > m) = P (X > n)$ .

With $p = 0.25$ , $P (X = 3) = (0.75)^{2} (0.25) = 0.140625$ .

OpenStax, Introductory Statistics 2e, "Chapter 4: Discrete Random Variables", https://openstax.org/books/introductory-statistics-2e/pages/4-introduction ↩︎