Geometric Distribution — Mathematics Knowledge System

Overview

The geometric distribution models the number of independent Bernoulli(p) trials needed to get the first success. In the common 'number of trials' formulation, X takes values 1, 2, 3, ... with PMF P(X=k) = (1-p)^{k-1}p, since the first k-1 trials must fail and the k-th must succeed. It is the discrete analog of the exponential distribution and is the unique discrete distribution with the memoryless property: given that the first success hasn't occurred by trial n, the additional number of trials needed still follows the same geometric distribution. Its mean is 1/p and its variance is (1-p)/p².

Intuition

Think of repeatedly flipping a biased coin until it lands heads for the first time. If heads is rare (p is small), you'd expect to wait a long time on average — and indeed E[X] = 1/p grows as p shrinks. The memoryless property captures something surprising: if you've already flipped 10 times without success, the number of ADDITIONAL flips needed still behaves exactly like a fresh geometric random variable — the coin has no 'memory' of your bad luck so far.

Formal Definition

Definition

For a sequence of independent Bernoulli(p) trials, let X be the trial number of the first success:

P(X = k) = (1-p)^{k-1}p, \qquad k = 1, 2, 3, \ldots

PMF

E[X] = \frac{1}{p}

Mean

\operatorname{Var}(X) = \frac{1-p}{p^2}

Variance

P(X > n) = (1-p)^n

Memoryless property (tail probability)

Worked Examples

Apply the geometric PMF with k=3, p=0.25.
$P(X=3) = (1-0.25)^{3-1}(0.25) = (0.75)^2(0.25)$
Compute the numeric value.
$P(X=3) = 0.5625 \times 0.25 = 0.140625$

Answer: P(X=3) = 0.140625 (about 14.1%).

Practice Problems

Difficulty 3/10

A fair coin (p=0.5) is flipped repeatedly. Find P(first heads occurs on flip 4).

Difficulty 4/10

A machine part fails an inspection with probability p=0.1 each test. Find the expected number of tests until the first failure.

Difficulty 5/10

A sales rep closes a deal with probability 0.2 per call, independently. Using the memoryless property, what is the probability the rep needs MORE than 5 additional calls to close a deal, given they've already made 3 calls with no success?

Quiz

Summary

Geometric(p) models the trial number of the first success in repeated independent Bernoulli(p) trials.
PMF: P(X=k) = (1-p)^{k-1}p; mean 1/p; variance (1-p)/p².
It is the unique discrete memoryless distribution — past failures don't change the distribution of future trials needed.

References

WebsiteWikipedia — Geometric distribution