Question

x1 and x2 are indep. random variables with parameter equal to λ1 and λ2. show that sum of x1 and x2 is also a poisson random variable.

Answer 1

cant remember exactly, but try to use characteristic function

Answer 2

need to use moment generating function approach unfortunatley

Answer 3

\[P(X1+X2=k)=\sum_{m}^{?}P(X1=m)P(X2=k-m)\]=\[\sum_{m=0}^{k}\lambda^m1*e^(-\lambda1)/(m!) *\lambda^m2*e^(-\lambda2)/((k-m)!) \]
=\[e^(\lambda1+\lambda2)/k!\sum_{m=0}^{k}k!/(m!(k-m)!*\lambda^m1\lambda^(k-m)2\]=\[e^-(\lambda1+\lambda2) /k! *(\lambda1+\lambda2)^k\]