Birth processes and the Poisson process

Flies and wasps land on your dinner plate in the manner of independent Poisson processes with respective intensities $\lambda$ and $\mu$. Show that the arrivals of flying objects form a Poisson process with intensity $\lambda +\mu$.

Insects land in the soup in the manner of a Poisson process with intensity $\lambda$, and each such insect is green with probability $p$, independently of the colours of all other insects. Show that the arrivals of green insects form a Poisson process with intensity $\lambda p$.

Let $T_{n}$ be the time of the $n$th arrival in a Poisson process $N$ with intensity $\lambda$, and define the excess lifetime process $E(t) = T_{N(t)+1}-t$, being the time one must wait subsequent to $t$ before the next arrival. Show by conditioning on $T_{1}$ that

Solve this integral equation in order to find the distribution function of $E(t)$. Explain your conclusion.

Let $B$ be a simple birth process of paragraph $(6.8.15 \mathrm{~ b})$ with $B(0) = I$; the birth rates are $\lambda_{n} = n \lambda$. Write down the forward system of equations for the process and deduce that

Show also that $\mathbb {E}\left[B(t)\right] = I e^{\lambda t}$ and $\operatorname {Var}\left[B(t)\right] = I e^{2 \lambda t}\left(1-e^{-\lambda t}\right)$.

Let $B$ be a process of simple birth with immigration (6.8.11c) with parameters $\lambda$ and $\nu$, and with $B(0) = 0$; the birth rates are $\lambda_{n} = n \lambda +\nu$. Write down the sequence of differential-difference equations for $p_{n}(t) = \mathbb {P}\left(B(t) = n\right)$. Without solving these equations, use them to show that $m(t) = \mathbb {E}\left[B(t)\right]$ satisfies $m^{\prime }(t) = \lambda m(t)+\nu$, and solve for $m(t)$.

Let $N$ be a birth process with intensities $\lambda_{0}, \lambda_{1}, \ldots$, and let $N(0) = 0$. Show that $p_{n}(t) = \mathbb {P}\left(N(t) = n\right)$ is given by

provided that $\lambda_{i} \neq \lambda_{j}$ whenever $i \neq j$.

Suppose that the general birth process of the previous exercise is such that $\sum_{n} \lambda_{n}^{-1} < \infty$. Show that $\lambda_{n} p_{n}(t) \rightarrow f(t)$ as $n \rightarrow \infty$ where $f$ is the density function of the random variable $T = \sup \left\{ t: N(t) < \infty \right\}$. Deduce that $\mathbb {E}\left[N(t) \mid N(t) < \infty \right]$ is finite or infinite depending on the convergence or divergence of $\sum_{n} n \lambda_{n}^{-1}$.

Find the Laplace transform of $f$ in closed form for the case when $\lambda_{n} = \left(n+\frac{1}{2}\right)^{2}$, and deduce an expression for $f$.

A traffic light is green at time $0$, and subsequently alternates between green and red at the instants of a Poisson process with intensity $\lambda$. Starting from time $x > 0$, let $W(x)$ be the waiting time until the light is green for the first time. Find the distribution of $W(x)$.

Conditional property of simple birth. Let $X = \left\{ X(t): t \geq 0\right\}$ be a simple birth process with rate $\lambda$, and let $X(0) = 1$. Let $b \geq 1$. Show that, conditional on the event $\left\{ X(t) = b+1\right\}$, the times of the $b$ births have the same distribution as the order statistics of a random sample of size $b$ from the density function

Boulders fall down a chute (or couloir) at the instants of a Poisson process with intensity $\lambda$, and mountaineers ascend the chute at the instants of a Poisson process with intensity $\mu$ (the two processes are independent of one another). If a fall and an ascent occur during any interval of length $c$ or less, it is said that a coincidence has occurred. Show that the time $T$ until the first coincidence has mean

Show that $c \mathbb {E}\left[T\right] \rightarrow (2 \lambda \mu )^{-1}$ as $c \downarrow 0$. Can you prove the last result directly?

Let $S_{1}, S_{2}$ be the times of the first two arrivals in a Poisson process with intensity $\lambda$, in order of arrival. Show that

and deduce the joint density function of $S_{1}$ and $S_{2}$.

A freelance salesman is paid $R$ units for each sale, and commissions arrive at the instants of a Poisson process of rate $\lambda$. Living costs consume his resources at unit rate. If his initial wealth is $S > 0$, show that the probability that he ever becomes bankrupt is $a^{S / R}$, where $a$ is the smallest $x > 0$ such that $x = e^{\lambda R(x-1)}$.

Prove that $a \in (0,1)$ if $\lambda R > 1$, while $a = 1$ if $\lambda R < 1$.