The geometric distribution is a "waiting" distribution. It arises when a sequence of independent Bernoulli trials are held, each with the same probability of success p. The trial at which the first success occurs is a geometric(p) random variable X. Because the earliest this can occur is the first trial, geometric random variables can only take positive integer values.
| Parameter | Range | Description |
|---|---|---|
| p | 0 < p ≤ 1 | Probability of success |
Probability Mass Function
Support
Mean
Variance
| Example | p |
|---|---|
| A fair coin is tossed repeatedly. Let X be the first toss at which a head occurs. | 0.5000 |
| When calling a customer support line, the probability of speaking to a human being in the first minute is 0.2. In repeated calls, let X be the first call at which this occurs. | 0.2000 |
| When using a web dating site, the probability that an initial date leads to a second one is 10%. Let X be the first date which leads to a second one. | 0.1000 |
X ~ Geometric(p)
E(X) = , Var(X) =
A first success on trial x means there are x - 1 failures followed by one success. Probabilities for this distribution therefore follow a geometric sequence with ratio 1 - p, since each failure has probability 1 - p.
The illustration above shows a sequence of independent Bernoulli(p) trials. The trial at which the first success occurs has a geometric(p) distribution.
The simulation above shows a sequence of independent Bernoulli(p) trials. Successful trials are shown in green and failures in grey. The random variable X is the trial at which the first success occurs, which has a geometric(p) distribution. The histogram accumulates the results of each simulation.