Geometric Distribution

Waiting time to the first success: mass function, survival, and mean.

The idea

The geometric distribution models the number of trials needed to obtain a first success. Run independent trials, each succeeding with probability $p$, and let $X$ be the number of the trial on which the first success occurs. Any number of failures in a row is possible, so $X$ can take any of the values $1, 2, 3, \ldots$ — infinitely many.

To find $\mathbb{P}(X = n)$, observe that exactly one history puts the first success on trial $n$: the first $n - 1$ trials all fail, and the $n$-th succeeds. The trials are independent, so these probabilities multiply: $\mathbb{P}(X = n) = (1-p)^{n-1}\,p, \qquad n = 1, 2, 3, \ldots$ Each additional trial of waiting contributes one more factor of $1 - p$, so consecutive probabilities stand in the constant ratio $1 - p$. A sequence with a constant ratio is called geometric, and the distribution takes its name from this.

Ways to work on it

• Walkthrough. The mass function, the survival probability, and the mean.
• Practice. Evaluate the geometric mass function.
• Hardest. Survival probability and the memoryless property.

Not sure where to start? Take the ten-question placement test.