P&S: Poisson and Geometric Distributions

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What is a Poisson distribution?

Back

A Poisson distribution is a probability distribution that expresses the probability of a given 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 a Geometric distribution?

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A Geometric distribution is a probability distribution that models the number of trials needed to get the first success in a series of independent Bernoulli trials, where each trial has the same probability of success.

Front

What is the formula for the Poisson probability mass function?

Back

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

Front

What is the formula for the Geometric probability mass function?

Back

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

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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.

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What does the parameter λ represent in a Poisson distribution?

Back

The parameter λ (lambda) represents the average number of events in a fixed interval of time or space.

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What is the expected value (mean) of a Poisson distribution?

Back

The expected value (mean) of a Poisson distribution is equal to λ.

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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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What is the relationship between the Poisson and Exponential distributions?

Back

The Poisson distribution models the number of events in a fixed interval, while the Exponential distribution models the time between events in a Poisson process.

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How do you find the probability of exactly k events occurring in a Poisson distribution?

Back

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