Calculate the covariance between the values X(t),X(t+s) taken by piossion process X(t) with constant lambda at times t and t+s, where s>0 i am having problem in the solution: how come Cov(X(t),X(t+s)) = Cov(X(t), X(t+s)+(x(t+s)- X(t))) please clarify regards Suresh sharma
sorry not clear . as per covariance property Cov(x,(x+y))= Cov(x,x)+cov(x,y) so we could have done Cov(x(t),x(t+s))= Cov(x(t),x(t))+Cov(X(t),X(s)) please clarify
cov(x,x) = var(x) and cov( x(t), x(t+s) -x(t))=0 since increments are independent of the past in a poisson process. now var (x(t))= lambda*t since x(t) is a poisson variable with parametrer (lambda*t) .