(a) Define the Poisson process $\left(N_{t}, t \geqslant 0\right)$ with rate $\lambda>0$, in terms of its holding times. Show that for all times $t \geqslant 0, N_{t}$ has a Poisson distribution, with a parameter which you should specify.

(b) Let $X$ be a random variable with probability density function

$$f(x)=\frac{1}{2} \lambda^{3} x^{2} e^{-\lambda x} \mathbf{1}_{\{x>0\}} .$$

Prove that $X$ is distributed as the sum $Y_{1}+Y_{2}+Y_{3}$ of three independent exponential random variables of rate $\lambda$. Calculate the expectation, variance and moment generating function of $X$.

Consider a renewal process $\left(X_{t}, t \geqslant 0\right)$ with holding times having density $(*)$. Prove that the renewal function $m(t)=\mathbb{E}\left(X_{t}\right)$ has the form

$$m(t)=\frac{\lambda t}{3}-\frac{1}{3} p_{1}(t)-\frac{2}{3} p_{2}(t)$$

where $p_{1}(t)=\mathbb{P}\left(N_{t}=1 \bmod 3\right), p_{2}(t)=\mathbb{P}\left(N_{t}=2 \bmod 3\right)$ and $\left(N_{t}, t \geqslant 0\right)$ is the Poisson process of rate $\lambda$.

(c) Consider the delayed renewal process $\left(X_{t}^{\mathrm{D}}, t \geqslant 0\right)$ with holding times $S_{1}^{\mathrm{D}}, S_{2}, S_{3}, \ldots$ where $\left(S_{n}, n \geqslant 1\right)$, are the holding times of $\left(X_{t}, t \geqslant 0\right)$ from (b). Specify the distribution of $S_{1}^{\mathrm{D}}$ for which the delayed process becomes the renewal process in equilibrium.

[You may use theorems from the course provided that you state them clearly.]