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The geometric distribution counts the number of Bernoulli trials until a success is achieved, with a fixed probability of success and failure. It is named after the geometric series of its successive terms and has the no-memory property. The mean is 1/p and the variance is q/p2.
The geometric distribution is a discrete probability distribution that counts the number of Bernoulli trials until a success is achieved. A Bernoulli process is an independent repeatable event with a fixed probability p of success and probability q=1-p of failure, such as a coin toss. Examples of variables with a geometric distribution include counting the number of times a pair of dice must be rolled until a 7 or 11 is rolled, or examining products on an assembly line until a defect.
This is called the geometric distribution because its successive terms form a geometric series. The probability of success on the first trial is p, the probability on the second trial is pq, the probability on the third trial is pq2, and so on. The generalized probability for the nth term is pqn-1 which is the probability of n-1 consecutive failures multiplied by the probability of success in the final test. The geometric distribution is a specific example of a negative binomial distribution that counts the number of Bernoulli trials until r successes are obtained. Some texts also call it the Pascal distribution, although others use the term more generally for any negative binomial distribution.
The geometric distribution is the only discrete probability distribution with the no-memory property, which states that the probability is unaffected by what has come before. This is a consequence of the independence of the Bernoulli trials. For example, if the variable is the number of times a roulette wheel must be spun to turn black, the number of times the wheel has turned red before the count begins does not affect the distribution.
The mean of a geometric distribution is 1/p. Thus, if the probability that a product on the assembly line is defective is 0025, one would expect to examine an average of 400 products before finding a defect. The variance of a geometric distribution is q/p2.
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