Compute $E[N_t N_s]$ function of Poisson Process

Question

Let $N_t$ be a Poisson process with parameter $\lambda$. Compute the function:

$$c(s,t) = E[N_t N_s]$$

My attempt:

Write: $E[N_t N_s] = E[E[N_t N_s|N_s]]=E[N_s E[N_t|N_s]]=E[N_sE[N_{|t-s|}]]=E[N_s]E[N_{|t-s|}] = (s\lambda) * (|t-s| \lambda) = \lambda^2s|t-s|$

I have a feeling I have a mistake due to lack of symmetry between $s$ and $t$ in the final result. Am I making any mistakes in the computation?

Answer 1

Assuming $t<s$, thinking in terms of disjoint blocks of time gives \begin{align*} E[N_t N_s]&= E[N_t\{N_s-N_t+N_t\}]\\ &=E[N_t]E[N_s-N_t]+E[N_t^2]\\ &=\lambda t(s-t)\lambda+\text{Var}[N_t]+E^2[N_t]\\ &=\lambda t(s-t)\lambda+\lambda t+(\lambda t)^2. \end{align*}