Geometric and Poisson Distributions

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What is a Geometric Distribution?

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

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The probability mass function is given by 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 a Poisson Distribution?

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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 Poisson Distribution?

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The probability mass function is given by 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 determine the probability of the first success occurring on the k-th trial in a Geometric Distribution?

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You use the formula P(X = k) = (1 - p)^(k-1) * p, where p is the probability of success.

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What is the mean of a Geometric Distribution?

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The mean of a Geometric Distribution is 1/p, where p is the probability of success.

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What is the variance of a Geometric Distribution?

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The variance of a Geometric Distribution is (1 - p) / p^2.

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What is the mean of a Poisson Distribution?

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The mean of a Poisson Distribution is equal to λ, the average rate of occurrence.

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What is the variance of a Poisson Distribution?

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The variance of a Poisson Distribution is also equal to λ.

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

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You use the formula P(X = k) = (λ^k * e^(-λ)) / k!.