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.
The function ⌊x⌋ in the cdf denotes the "floor" or greatest integer function.
| 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.
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.
Y =
E(Y) = , Var(Y) =