Question

Consider the random sum S= Xj, where the X, are IID Bernoulli random variables with parameter p and N is a Poisson random variable with parameter 1. N is independent of the X; values. a. Calculate the MGF of S. b. Show S is Poisson with parameter Ap. Here is one interpretation of this result: If the number of people with a certain disease is Poisson with parameter 1 and each person tests positive for the disease with probability p, then the number of people who test positive is Poisson with parameter Ap.

Answer 1

N MGF f *: : My (8) - Eletxt) - {et* P(xi-x): <HO:(1+4)+ e* p = (2+)+tet). a MGF&x: Mg(t) - Elets)- E(EC+SIN=n)) [ homof iterated expectation] = E(Elet: Lex In:n)) - ECE[et]<:)) = E(T 7(ex) [:: xisome id] = E ((-+)+ket)" [inan (nica redization f m) ] • E(Q+)+ket) Na mo [- Nu Poimen(t) ] P(n=1) Set I (02) [a=Q+)rbet] [es ] não n 2-0 2 اے د ۴۰ دے ، ا ا د . (-) د : عوده : by MGF of Poisson (x) . Let Yn Poisson (it) : My fé) - Ele+x)= **PCY+4)= .54 com cv E Cadety a e the date [ie ] ed idet By uniqueness of MGF, S~ Poisson (sip) 22 241