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Geometric distribution is a probability distribution that models the number of trials needed to achieve a success.

Geometric distribution is a discrete probability distribution that models the number of independent trials needed to achieve a success in a Bernoulli trial. A Bernoulli trial is a random experiment with two possible outcomes, success or failure, and the probability of success is denoted by p. The geometric distribution is often used to model situations where we are interested in the number of trials needed to achieve a success, such as the number of times we need to flip a coin until we get a head.

The probability mass function (PMF) of the geometric distribution is given by:

P(X=k) = (1-p)^(k-1) * p

where X is the random variable representing the number of trials needed to achieve a success, k is a positive integer representing the number of trials, and p is the probability of success.

The mean and variance of the geometric distribution are given by:

E(X) = 1/p

Var(X) = (1-p)/p^2

The geometric distribution is memoryless, which means that the probability of achieving a success on the next trial is the same regardless of how many trials have already been performed. This property makes the geometric distribution useful in modelling situations where the probability of success remains constant over time, such as in radioactive decay or machine failure.

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