Question

Let \(\{N(t),t\geq 0\}\) be a Poisson process with rate \(\lambda=1\). Then \(E[N(8)N(5)]\) equals (in integer).

Hint:

For a Poisson process, increments over disjoint intervals are independent. Always split \(N(t)\) as \(N(s)+[N(t)-N(s)]\) when \(t>s\).

Answer 1

Correct Answer: 45

Step 1: Split the larger count.
Write $N(8)=N(5)+M$ where $M=N(8)-N(5)$ is independent of $N(5)$. With rate $1$, $N(5)\sim$ Poisson$(5)$ and $M\sim$ Poisson$(3)$.

Step 2: Expand the product.
$E[N(8)N(5)]=E[N(5)^2]+E[M\,N(5)]$.

Step 3: First piece.
$E[N(5)^2]=\mathrm{Var}+\text{mean}^2=5+25=30$.

Step 4: Second piece.
By independence $E[M\,N(5)]=3\cdot5=15$.

Step 5: Add.
$30+15=45$.
\[ \boxed{45} \]