# Poisson distribution probabilities and recursion relationship

The Poisson distribution is given by the following

P(x,Î»)=e ^ -Î» * Î»^x! x=0,1,2,3.....j.....

Where Î»>0 is a parameter which is the average value μ in poisson distribution.

a) show that the maximum poisson probability P(x=j,Î») occurs at approximately the average value, that is Î»=j if Î»>1.

(hint: you can take the first order derivative of the natural log of poisson probability, P(x=j, Î») with respect to Î» and set it equal to 0

b) show that when Î»<1 the poisson probability is a monotonically decreasing function of j, i.e, P(0, Î»)>P(1, Î»)>P(2, Î»)....P(j, Î»).... And never has a maximum value

(hint: you can use the recursion relationship of the poisson distribution to prove this statement)

https://brainmass.com/statistics/probability/poisson-distribution-probabilities-recursion-relationship-279724

#### Solution Preview

The Poisson distribution is given by the following

P(x,Î»)=e ^ -Î» * Î»^x/x! x=0,1,2,3.....j.....

Where Î»>0 is a parameter which is the average value Î¼ in poisson distribution.

a) show that the maximum poisson probability P(x=j,Î») occurs at approximately the average value, that is Î»=j if ...

#### Solution Summary

Poisson distribution probabilities and recursion relationships are examined.