Pure birth processes

Show that a simple birth process $X$ with parameter $\lambda$ is a doubly stochastic Poisson process with intensity function $\lambda (t) = \lambda X(t)$.

The Markov chain $X = \left\{ X(t): t \geq 0\right\}$ is a birth process whose intensities $\lambda_{k}(t)$ depend also on the time $t$ and are given by

as $h \downarrow 0$. Show that the probability generating function $G(s, t) = \mathbb {E}\left[s^{X(t)}\right]$ satisfies

Hence find the mean and variance of $X(t)$ when $X(0) = I$.

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 $f$.

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

Assuming that in the reproduction of bacteria by division (into two bacteria), the probability of a bacterium dividing during a time interval $\Delta$ is $\lambda \Delta + o(\Delta )$ as $\Delta \to 0$ and does not depend on the number of previous divisions or on the number of bacteria present, find the probability that, if there was one bacterium at time 0, there will be $i$ bacteria at time $t$.