Geometric and Poisson Distributions

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What is a Geometric Distribution?

Back

A Geometric Distribution models the number of trials until the first success in a series of independent Bernoulli trials, where each trial has the same probability of success.

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What is a Poisson Distribution?

Back

A Poisson Distribution models the number of events occurring in a fixed interval of time or space, given that these events happen with a known constant mean rate and independently of the time since the last event.

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What is the formula for the probability mass function of a Geometric Distribution?

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

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What is the formula for the probability mass function of a Poisson Distribution?

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P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the average rate of occurrence and k is the number of occurrences.

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How do you calculate the mean of a Geometric Distribution?

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Mean = 1/p, where p is the probability of success.

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How do you calculate the mean of a Poisson Distribution?

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Mean = λ, where λ is the average rate of occurrence.

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What is the variance of a Geometric Distribution?

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Variance = (1-p) / p^2.

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What is the variance of a Poisson Distribution?

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Variance = λ, where λ is the average rate of occurrence.

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If the average number of shark attacks is 10 per year, what is the probability of exactly 2 attacks next year?

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P(X = 2) = (10^2 * e^(-10)) / 2! = 0.0028.

Front

What is the probability that the first success occurs on the fourth trial in a Geometric Distribution with p = 0.23?

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P(X = 4) = (1-0.23)^(4-1) * 0.23 = 0.1859.