Poisson process independence

Question

In a Poisson process with rate $\lambda$ we know that the number of arrivals from $t = 0$ to $t = 1$ is independent of the number arrivals $t = 1$ to $t = 2$. However, are $N(2)$ and $N(1)$ independent? They have overlapping time intervals, which would seem to indicate that they are not...

Answer 1

No, $N(1)$ and $N(2)$ are not independent. We have

$$\mathbb{E}N(t) = \lambda \cdot t \qquad \qquad \text{var} \, N(t):=\mathbb{E}[(N(t)-\mathbb{E}N(t))^2]=\lambda t \tag{1}$$

for any $t \geq 0$ as $(N(t))_{t \geq 0}$ is a Poisson process. In particular,

$$\mathbb{E}N(1) \cdot \mathbb{E}N(2) \stackrel{(1)}{=} (\lambda \cdot 1) \cdot (\lambda \cdot 2)=2\lambda^2$$

On the other hand,

$$\begin{align*} \mathbb{E}[N(1) \cdot N(2)] &= \mathbb{E}[N(1) \cdot (N(2)-N(1))] + \mathbb{E}[N(1)^2] \\ &= \underbrace{\mathbb{E}[N(2)-N(1)] \cdot \mathbb{E}N(1)}_{[\mathbb{E}N(1)]^2} + \mathbb{E}(N(1)^2) \\ &= \text{var} \, N(1)+2 [\mathbb{E}N(1)]^2 \stackrel{(1)}{=} \lambda+2\lambda^2 \end{align*}$$

where we used that $N_2-N_1$ and $N_1$ are independent and $N_2-N_1 \sim N_{2-1} = N_1$; by the stationarity and independence of the increments. This shows that

$$\mathbb{E}(N(1) \cdot N(2)) \neq \mathbb{E}N(1) \cdot \mathbb{E}N(2).$$

Consequently, $N(1)$ and $N(2)$ are not independent.