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.
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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.
The mean (expected value) of a geometrically distributed random variable is $E(X) = \frac{1}{p}$.
The variance of a geometrically distributed random variable is $Var(X) = \frac{1-p}{p^2}$.
Geometric distributions assume that each trial is independent and has only two possible outcomes: success or failure.
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?
A probability distribution used to describe the number of events occurring within a fixed interval, with events happening independently at a constant rate.