Geometric Distribution Calculator
Geometric distribution calculator. Compute probability of first success on the k-th trial or number of failures before first success, plus CDF, mean, and variance. Includes formulas and explanations.
Full original guide (expanded)
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.