Geometric Distribution Calculator

Compute probabilities for the geometric distribution in both common parameterizations: either the trial on which the first success occurs (k = 1, 2, 3, …) or the number of failures before the first success (x = 0, 1, 2, …). Enter the success probability p and the trial/failure number to get PMF, CDF, and tail probabilities.

0 < p ≤ 1

for trials: k ≥ 1

Showing first 10 probabilities for current parameterization.

PMF

—

CDF

—

Mean

—

E[X] = 1/p

Variance

—

Var = (1-p)/p²

Range result

—

k/x PMF CDF

Formulas

Parameterization 1 (trial of first success):

\( P(X = k) = (1 - p)^{k - 1} \, p \quad \text{for } k = 1, 2, 3, \dots \)

\( F(k) = P(X \le k) = 1 - (1 - p)^k \)

\( \mathbb{E}[X] = \frac{1}{p}, \quad \mathrm{Var}(X) = \frac{1 - p}{p^2} \)

Parameterization 2 (failures before first success):

\( P(Y = x) = (1 - p)^x \, p \quad \text{for } x = 0, 1, 2, \dots \)

\( F(x) = P(Y \le x) = 1 - (1 - p)^{x+1} \)

\( \mathbb{E}[Y] = \frac{1 - p}{p}, \quad \mathrm{Var}(Y) = \frac{1 - p}{p^2} \)

When to use the geometric distribution

  • Bernoulli trials, all independent.
  • Same success probability p at each trial.
  • Interested in the first success only.