Geometric distribution

Definition

A geometric distribution models the number of trials needed to get the first success in a series of independent and identically distributed Bernoulli trials. The probability of success remains constant across all trials.

5 Must Know Facts For Your Next Test

1. The probability mass function (PMF) is given by $P(X = k) = (1-p)^{k-1} p$ where $p$ is the probability of success, and $k$ is the trial number on which the first success occurs.
2. The mean (expected value) of a geometrically distributed random variable is $E(X) = \frac{1}{p}$.
3. The variance of a geometrically distributed random variable is $Var(X) = \frac{1-p}{p^2}$.
4. Geometric distributions assume that each trial is independent and has only two possible outcomes: success or failure.
5. The cumulative distribution function (CDF) for a geometric distribution can be expressed as $P(X \leq k) = 1 - (1-p)^k$.

Review Questions

• What does the geometric distribution model?
• How do you calculate the mean and variance for a geometrically distributed random variable?
• What are the assumptions required for a geometric distribution to be applicable?

Related terms

Bernoulli Trial: A random experiment where there are only two possible outcomes: success or failure.

Binomial Distribution: A probability distribution summarizing the likelihood that a given number of successes will occur in a fixed number of Bernoulli trials.

Poisson Distribution: A probability distribution used to describe the number of events occurring within a fixed interval, with events happening independently at a constant rate.