# Expectation of arrival times

Let $$(N_t)_t$$ be a Poisson process with parameter λ = 2. By $$τ_k$$ denote the time of the k-th arrival (k = 1, 2, . . .), and by $$ρ_k = τ_k −τ_{k−1}$$ - the interarrival time between the (k−1)th and kth arrival (k = 1, 2, . . .), with $$τ_0 = 0$$ (as in the construction of Poisson process).

Find the following:

(a) $$E(N_3N_4)$$
(b) $$E(ρ_3ρ_4)$$
(c) $$E(τ_3τ_4)$$

I can understand (a), and (b). But, couldn't understand (c).

I collected the following from a friend:
(c) $$E(τ_3τ_4) \\ = E[τ_3(τ_3+ρ_4)] \\ = E(τ^2_3 + ρ_4) \\ = E(τ^2_3) + E(ρ_4) \\ = (k/\lambda)^2 + 1/ \lambda$$

First of all, is this a correct solution? If not, what is the correct one?

The processing of $$τ_3$$ doesn't look okay to me.

It seems to me that $$ρ_4$$ is introduced here because $$τ_3$$ and $$τ_4$$ are not independent. Why are not they independent?

Suppose, the 3rd item arrived at 9:05 AM and the 4th one arrived at 9:10 AM. If the process started at 9:00 AM, the 3rd one took 5 minutes and the 4th one took 10 minutes to arrive. How are they not independent?

$$\tau_3$$ and $$\tau_4$$ are not independent because these are arrival times and they proceed additively, i.e. $$\tau_4=\tau_3+\rho$$. For example, if you know that $$\tau_3=20$$, it means $$\tau_4\geq20$$.
$$\tau_3$$ is Gamma distributed with $$(3,\lambda)$$ because it is sum of $$\rho_1,\rho_2,\rho_3$$, which are iid exponentially distributed RVs. In general sum of $$\alpha$$ iid exponentials is Gamma with $$(\alpha,\lambda)$$. For a Gamma RV, we have $$E[X^2]=\frac{k+k^2}{\lambda^2}$$, so here $$E[\tau_3^2]=\frac{12}{\lambda^2}$$.
• What are $N_3$ and $N_4$ in this context? I mean what are their relationships with $\tau_3$ and $\tau_4$? Apr 14, 2019 at 19:21
• What is $t$'s relationship with $\tau$? Apr 14, 2019 at 19:27
• Speaking generally, $N_t$ refers to number of arrivals which occur by time t. t is just a variable, just like k. And, it's not a random variables. You shouldn't seek for t's relation to $\tau$. Apr 14, 2019 at 19:29
• So, $t$ is just an index, like what we use in programming arrays? Apr 14, 2019 at 19:37
• similar, except that it is continuous. One can say $N_{3.512}$ for example, which will mean the number of events by $t=3.512$. Apr 14, 2019 at 19:38