I can't fully understand how to solve this question:
Let $\{N(t); t>0\}$ be a Poisson Process with intensity $\lambda$ > $0$. Compute $E(N(2)(N(4)-N(1)))$.
I should end up with
$E((N(2)−N(1))(N(4)−N(2)))+E((N(2)−N(1))^2)+E(N(1)(N(4)−N(1)))=2\lambda^2+\lambda+\lambda^2+3\lambda^2$,
but I can't figure out the reasoning. What I've come up with is:
$E((N(2)−N(1))(N(4)−N(2)))=E(N(2)-N(1))E(N(4)-N(2))=\lambda*2\lambda=2\lambda^2$
$E((N(2)−N(1))^2)=E(N(2)-N(1))E(N(2)-N(1))=\lambda^2$
$E(N(1)(N(4)−N(1)))=E(N(1))E(N(4)-N(1))=3\lambda^2$
Which of course don't add upp to the correct answer. Were are my reasonings wrong?