ch 2 stochastic process exam style question 2.12

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

 

Cov(X(t),X(t+s)) = Cov(X(t), X(t)+(X(t+s)- X(t))). now go forward and you are done.

 

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) .

 

No. This is not true, since X(t+s) is not equal to X(t)+X(s).

X(t+s) = X(t) + (X(t+s) - X(t))

 

thanks now its clear

 

thanks got it