Non-homogeneous Poisson processes

Let $N$ be a non-homogeneous Poisson process on $\mathbb {R}_{+}$with intensity function $\lambda$. Find the joint density of the first two inter-event times, and deduce that they are not in general independent.

Let $N$ be a non-homogeneous Poisson process on $\mathbb {R}_{+} = [0, \infty )$ with intensity function $\lambda$.

Successive offers for my house are independent identically distributed random variables $X_{1}$, $X_{2}, \ldots$, having density function $f$ and distribution function $F$. Let $Y_{1} = X_{1}$, let $Y_{2}$ be the first offer exceeding $Y_{1}$, and generally let $Y_{n+1}$ be the first offer exceeding $Y_{n}$. Show that $Y_{1}, Y_{2}, \ldots$ are the times of arrivals in a non-homogeneous Poisson process with intensity function $\lambda (t) = f(t) /(1-F(t))$. The $Y_{i}$ are called 'record values'.

Now let $Z_{1}$ be the first offer received which is the second largest to date, and let $Z_{2}$ be the second such offer, and so on. Show that the $Z_{i}$ are the arrival times of a non-homogeneous Poisson process with intensity function $\lambda$.

Consider a Poisson process with variable intensity, i.e., the probability of a jump occurring in the time interval $(t, t+h)$ does not depend on the past and equals $\lambda (t)h + o(h)$ as $h \to 0$ (note that $\lambda$ may depend on $t$).

Let $\left\{ X(t)\right\}$ be an inhomogeneous Poisson process with time dependent intensity $\lambda (t) = \gamma t$. Find, with $T_{1}$ as in the previous problem, the probabilities $\mathbb {P}\left(T_{1} \leq s\right)$ and $\mathbb {P}\left(T_{1} \leq s \mid X(t)=1\right)$.

The risk for car accidents on a specific road varies with the time of the day; denote by $\lambda (t)$ the intensity with which there occurs an accident at a certain road at time $t$, $0 \leq t \leq 24$.