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## Homework Statement

A random variable has a Poisson distribution with parameter λ = 2. Compute the following probabilities, giving an exact answer and a decimal approximation.

P(X ≥ 4)

## Homework Equations

P(X = k) = λ

^{k}e

^{-λ}/k!

## The Attempt at a Solution

P(X ≥ 4) = Ʃ

_{k = 4}

^{∞}λ

^{k}e

^{-λ}/k!

= λ

^{4}e

^{-λ}/4! + λ

^{5}e

^{-λ}/5! + λ

^{6}e

^{-λ}/6! + [itex]\cdots[/itex]

= e

^{-λ}[λ

^{4}/4! + λ

^{5}/5! + λ

^{6}/6! + [itex]\cdots[/itex]]

= e

^{-λ}Ʃ

_{k = 4}

^{∞}λ

^{k}/k!

Let n = k - 4

=e

^{-λ}Ʃ

_{n = 0}

^{∞}λ

^{n+4}/(n+4)!

plug in λ = 2

= e

^{-2}Ʃ

_{n = 0}

^{∞}2

^{n+4}/(n+4)!

= e

^{-2}Ʃ

_{n = 0}

^{∞}2

^{n}2

^{4}/(n+4)!

= 16e

^{-2}Ʃ

_{n = 0}

^{∞}2

^{n}/(n+4)!

This is as far as I have gotten, but I'm not sure I'm on the correct track. I used wolfram alpha to reduce the summation term, Ʃ

_{n = 0}

^{∞}2

^{n}/(n+4)!, to [1/48(3e

^{2}-19)], but I'm at a loss as to how to get there myself.

Anyone have any suggestions? It would be greatly appreciated!