Poisson Probability Mass Function

P (X = x) = \frac{e ^{- λ} λ ^{x}}{x !}, E [X] = Var (X) = λ

Poisson counts rare events in a fixed interval. Mean and variance both equal λ. For large n and small p, Binomial(n,p) ≈ Poisson(λ=np). The Handbook also lists the Poisson recursive relation P(x+1) = λ P(x)/(x+1).

$λ$: Mean rate (events per interval)
$x$: Integer count 0,1,2,…
$P (X = x)$: Probability of exactly x events

Step-by-step solved example

Defects arrive as Poisson with λ = 2 per shift. Find P(X = 0) and P(X = 1).

Poisson counts are a discrete cousin of the normal for large λ.

1. Zero events
P(0) = e^{−2} ≈ 0.1353.
$P (0) = e^{- 2}$
2. One event
P(1) = e^{−2} · 2 / 1 = 2e^{−2} ≈ 0.2707.
$P (1) = 2 e^{- 2}$

Answer: P(0) ≈ 0.135, P(1) ≈ 0.271

10 practice questions

0/10 correct

1.For Poisson, E[X] equals

2.Var(X) for Poisson(λ) is

3.λ = 3, P(X=0) =

4.Binomial(n=100, p=0.02) is often approximated by Poisson with λ =

5.P(X=x+1)/P(X=x) equals

6.λ = 1, P(X=2) is nearest

7.A Poisson random variable is

8.If λ doubles, P(X=0)

9.Standard deviation of Poisson(λ=16) is

10.Sum of independent Poisson(λ1) and Poisson(λ2) is