# Probability Theory: Poisson Distribution

## 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) = λke/k!

## The Attempt at a Solution

P(X ≥ 4) = Ʃk = 4 λke/k!

= λ4e/4! + λ5e/5! + λ6e/6! + $\cdots$

= e4/4! + λ5/5! + λ6/6! + $\cdots$]

= e Ʃk = 4 λk/k!

Let n = k - 4

=e Ʃn = 0 λn+4/(n+4)!

plug in λ = 2

= e-2 Ʃn = 0 2n+4/(n+4)!

= e-2 Ʃn = 0 2n24/(n+4)!

= 16e-2 Ʃn = 0 2n/(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, Ʃn = 0 2n/(n+4)!, to [1/48(3e2-19)], but I'm at a loss as to how to get there myself.

Anyone have any suggestions? It would be greatly appreciated!

Ray Vickson
Homework Helper
Dearly Missed

## 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) = λke/k!

## The Attempt at a Solution

P(X ≥ 4) = Ʃk = 4 λke/k!

= λ4e/4! + λ5e/5! + λ6e/6! + $\cdots$

= e4/4! + λ5/5! + λ6/6! + $\cdots$]

= e Ʃk = 4 λk/k!

Let n = k - 4

=e Ʃn = 0 λn+4/(n+4)!

plug in λ = 2

= e-2 Ʃn = 0 2n+4/(n+4)!

= e-2 Ʃn = 0 2n24/(n+4)!

= 16e-2 Ʃn = 0 2n/(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, Ʃn = 0 2n/(n+4)!, to [1/48(3e2-19)], but I'm at a loss as to how to get there myself.

Anyone have any suggestions? It would be greatly appreciated!

##P(X \geq 4) = 1-P(X \leq 3).##

• 1 person
BruceW
Homework Helper
Ray's 'clue' is what you must use for this problem. generally, this would be a difficult problem. But luckily, 4 is not a large number.

Doh! That little fact completely slipped my mind. Thanks guys!