## anonymous 5 years ago 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.

1. mathmagician

cant remember exactly, but try to use characteristic function

2. anonymous

need to use moment generating function approach unfortunatley

3. mathmagician

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