Geometric Distribution

The Geometric distribution models waiting time until the first success in repeated independent Bernoulli trials.

• Each trial succeeds with probability $p$ (same $p$ every time)
• Trials are independent
• The random variable $X$ is the trial index of the first success ($X = 1,2,3,\dots$)

PMF

Intuition

Think of “keep trying until it works”. The probability shrinks geometrically as $k$ grows, because you must fail $k-1$ times in a row, then succeed.

When to use

Use Geometric when:

• You are counting how many attempts until the first success
• The success probability is constant each attempt
• Attempts are independent

Common examples: first sale, first defect, first conversion, first time an event occurs.

Key facts

• Mean: $E[X] = \frac{1}{p}$
• Variance: $\mathrm{Var}(X) = \frac{1-p}{p^2}$
• Memoryless property:

Meaning: once you’ve already waited $s$ trials with no success, your future waiting time distribution resets.

Common pitfall

Some books define $X$ as the number of failures before the first success (support $0,1,2,\dots$). In that parameterization:

This guide uses trial count (support starts at 1).

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