Question

Let M be a Poisson (λ) random variable having M equal m. If we flip a p-biased coin m times and let X be the number of heads, show that X is a Poisson (pλ) random variable. Use the identity for k= 0 to infinity Σy^k/k! =e^y

Answer 1

solution : we know *Mv poisson (1) (Xam) ~ Binomial (m, p) (%/mcm) are the pmf of m and PCM-m) = ēten im ; meo, L, 24-. 3 o ; otherwise and p(x=x/mem) =) im pl. (1-P)"-2 ; x = 0,1,2--M l o ; otherwise Now, we want to find morginal distribution of X :P(X=x) = 3 MM = E P(x=x, MFM) { " P(mcm) P(X=x / Mem) Leo Exam (m) pe cr-pm-x ma CI-P mel mi mm-2 m 2 é xix px ye a Ź max max cop) del mel (max)

- é? (pje to Caci-p) B! MEK (MMX)]. eº (AP)* (1-2) : E é 1 +2 -1p Cape R1 (xp)* = è XP OL = 0, 1, 2,-.. .:: PCX=ra, é tp aple PIXES LEO,1,2,-.. otherwise Therefore X w poisson cap)