Geometric & 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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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 occur 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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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 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 does the mean of a Geometric Distribution represent?

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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 does the mean of a Poisson Distribution represent?

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The mean of a Poisson Distribution is λ, which is the average number of events in the given interval.

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How do you find the probability of more than k events occurring in a Poisson Distribution?

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You can find this by calculating 1 - P(X ≤ k), where P(X ≤ k) is the cumulative probability of k or fewer events.

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

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

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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.