Geometric and Poisson Distributions

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What is a Geometric Distribution?

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A probability distribution that models the number of trials until the first success in a series of independent Bernoulli trials.

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What is a Poisson Distribution?

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A probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence.

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What is the mean of a Poisson Distribution?

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The mean (λ) of a Poisson distribution is the average number of occurrences in the interval.

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How do you calculate the probability of exactly k events in a Poisson Distribution?

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P(X = k) = (λ^k * e^(-λ)) / k! where λ is the mean number of events and e is Euler's number.

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What is the formula for the probability of success in a Geometric Distribution?

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P(X = k) = (1 - p)^(k-1) * p, where p is the probability of success on each trial.

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What is the variance of a Poisson Distribution?

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The variance of a Poisson distribution is equal to its mean (λ).

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What is the relationship between the Poisson and the Exponential Distribution?

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The time between events in a Poisson process follows an Exponential distribution.

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How do you find the probability of at least one event occurring in a Poisson Distribution?

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P(X ≥ 1) = 1 - P(X = 0) = 1 - (e^(-λ)).

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What is the expected value of a Geometric Distribution?

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The expected value (mean) of a Geometric distribution is 1/p, where p is the probability of success.

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What is the cumulative distribution function (CDF) for a Geometric Distribution?

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F(k) = 1 - (1 - p)^k, where k is the number of trials.