Let $\{X(t):t\ge 0\} $ be a Poisson process with rate $\lambda$. T is a positive constant. Define $\xi_T=T^{-1}\int_0^TX(t)dt$.
Calculate $Var[\xi_T]$.
In the beginning, I want to solve it by definition of variance, i.e. $Var(\xi_T)=E(\xi_T^2)-E^2(\xi_T)$. But I was stuck at the step of calculating $E(\xi_T^2)$. Then I find it very easy if I directly change the order of variance and integration. I want to know if the second method is true or not. Or how to calculate $E(\xi_T^2)$?