Geometric Distribution

Basics of Geometric Distribution

• The geometric distribution models the number of failures before the first success in a series of Bernoulli trials.
• It is a discrete probability distribution where each trial is independent of each other and each has two outcomes, referred to as ‘success’ and ‘failure’.
• The probability of success, denoted by p, remains constant for each trial.

Properties of Geometric Distribution

• The probability mass function (pmf) of a geometric distribution is given by P(X=x) = p(1-p)^(x-1) where ‘x’ is the number of trials up to and including the first success.
• The expected value (mean) of a geometric distribution is E(X) = 1/p. This explains that on average, 1/p trials are needed before the first success.
• The variance of a geometric distribution is Var(X) = (1-p) / p^2.

Applications of Geometric Distribution

• The geometric distribution is particularly useful in applications where we are interested in the number of trials needed to obtain the first success in a series of independent and identically distributed (i.i.d.) Bernoulli trials.
• Typical examples involve quality control testing in manufacturing processes or in modelling the number of attempts before a request to a server succeeds in computer science.

Key Points to Remember

• The geometric distribution models the number of failures before the first success in independent and identically distributed Bernoulli trials.
• The trials are called independent because the outcome on any given trial doesn’t affect the outcome on any other trial.
• The trials are called identically distributed because the probability of success remains constant from trial to trial.
• Memorylessness property is an important characteristic of geometric distribution. This means that the probability of an event happening in the future does not depend on the past events or outcomes.