# What are the mean and standard deviation of a probability density function given by Pr(X=k) = (9^ke^-9)/(k!)  for k in {0,1,2... oo}?

Nov 28, 2016

mean =9

sd=3

#### Explanation:

this pdf is a Poisson distribution

X~Po(lambda)

P(X=k)=(lambda^ke^-k)/(k!)," " kin{1,2,3,4,...oo}

In such cases the mean (E(X)) and variance (Var(X)) are given by

$E \left(X\right) = \lambda , \text{ } V a r \left(X\right) = \lambda$

in this case we have

P(X=k)=(9^ke^-9)/(k!)," " kin{1,2,3,4,...oo}

so$\text{ } E \left(X\right) = 9$

$V a r \left(X\right) = 9$

$s d = \sqrt{V} a r = \sqrt{9} = 3$