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For a poisson distribution where X ~ Pois(u), solve the questions below. Please show all work...

For a poisson distribution where X ~ Pois(u), solve the questions below. Please show all work and all steps.

a.) Show that the pmf is a pmf using the criteria for verifying pmf (2 conditions).

b.) Show that E(X) = u

c.) Show that Var(X) = u

Solutions

Expert Solution

Let ,

Therefore , the probability mass function of X is ,

; x=0,1,2,..........

= 0 ; otherwise

a) Now, we shall verify that P(X) is a probability mass function

(i) ; for all x

(ii)

; Since the term corresponding to x=0 is zero

; Since ,

c) The second raw moment is ,

; Since ,

; Since the term corresponding to x=0 and x=1 are zero

Therefore ,

$Var(X)=\mu_{2}^{'}-\mu_{1}^{'}^2=\mu^2 +\mu-\mu^2=\mu$

milcah answered 2 months ago

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