Geometric Distribution Calculator

Calculate PMF, CDF, mean and variance for the geometric distribution (trials until first success).

Inputs

Success Probability (p)

Trial number of first success (k ≥ 1)

P(X = k)

0.10290000

P(X ≤ k)

0.75990000

Mean (1/p)

3.3333

Variance ((1−p)/p²)

7.7778

Step by step

Values used
Success Probability (p) = 0.3000; Trial number of first success (k ≥ 1) = 4
Formula applied
P(X=k) = p × (1−p)^(k−1), for k = 1, 2, 3, ...
P(X = k)
= 0.10290000
P(X ≤ k)
= 0.75990000
Mean (1/p)
= 3.3333
Variance ((1−p)/p²)
= 7.7778

How it works

The geometric distribution models the number of independent Bernoulli trials needed to get the first success. If p is the probability of success on each trial, then X counts which trial yields the first success (X ≥ 1).

Formula

P(X=k) = p × (1−p)^(k−1), for k = 1, 2, 3, ...

p: Success probability per trial
k: Trial number of first success

Frequently Asked Questions

What is the memoryless property?

The geometric distribution is the only discrete distribution that is memoryless: no matter how many failures have occurred, the probability of success on the next trial remains p.

How does this differ from the binomial?

The binomial counts successes in a fixed number of trials. The geometric counts trials until the first success — the number of trials is the random variable, not the number of successes.

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