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
Compute at most / at least / between for the selected parameterization.
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.
Formula (LaTeX) + variables + units
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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): \( P(Y = x) = (1 - p)^x \, p \quad \text{for } x = 0, 1, 2, \dots \) \( F(x) = P(Y \le x) = 1 - (1 - p)^{x+1} \) \( \mathbb{E}[Y] = \frac{1 - p}{p}, \quad \mathrm{Var}(Y) = \frac{1 - p}{p^2} \)
- No variables provided in audit spec.
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.