Poisson Distributions

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What is a Poisson Distribution?

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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 the formula for the Poisson probability mass function?

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P(X = k) = (λ^k * e^(-λ)) / k! where λ is the average rate (mean) of occurrence, k is the number of occurrences, and e is Euler's number (approximately 2.71828).

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If the mean number of events (λ) is 3, what is the probability of observing exactly 2 events?

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Using the Poisson formula: P(X = 2) = (3^2 * e^(-3)) / 2! = 0.224.

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What does the parameter λ represent in a Poisson Distribution?

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The parameter λ (lambda) represents the average number of occurrences in a specified interval.

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How do you calculate the mean of a Poisson Distribution?

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The mean of a Poisson Distribution is equal to the parameter λ.

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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 Poisson Distribution?

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The variance of a Poisson Distribution is equal to λ.

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In a Poisson Distribution, what is the probability of observing zero events if λ = 4?

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P(X = 0) = (4^0 * e^(-4)) / 0! = e^(-4) ≈ 0.0183.

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How can the Poisson Distribution be used in real-life scenarios?

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It can model the number of emails received in an hour, the number of phone calls at a call center, or the number of accidents at an intersection.

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

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The CDF gives the probability that a Poisson random variable is less than or equal to a certain value k, calculated as P(X ≤ k) = Σ (λ^i * e^(-λ) / i!) for i = 0 to k.