The Poisson process and its relatives

Let $\left\{ X(t)\right\}$ be a Poisson process with intensity $\lambda > 0$. Since the mean of the process is $m_{X}(t) = \lambda t$ we can conclude that the Poisson process is not weakly stationary. Is the process $Y(t) = X(t) - \lambda t$ weakly stationary?

In a Poisson process one has observed four events in the interval $(0,4]$. Compute the probability that two occurred in $(0,1]$ and the remaining two in $(1,2]$.

Fires in passenger trains are assumed to occur according to a Poisson process. During the period January 1, 1986 to December 31, 1994, there were 67 fires in passenger trains in Sweden. The total distance traveled by passenger trains was $544 \times 10^{6}$ km. Estimate the intensity of passenger train fires (per train km), and construct a two-sided confidence interval with approximate confidence level 95. You may use the normal approximation.

A Geiger counter is used to measure radioactive decay. To estimate the activity in a radioactive sample one can do the following experiment. First, the natural background radiation is measured during a time period $T_{b}$ and then, during a time period $T_{s}$, one measures the sample plus background.

Assuming both background and sample radiation are Poisson processes, with intensities $\lambda_{b}$ and $\lambda_{s}$, respectively, explain how the sample activity can be estimated. Also, for a fixed total experiment time $T = T_{b}+T_{s}$, find the best choice of $T_{b}$ and $T_{s}$ that makes the variance of the estimate as small as possible. Finally, construct an approximate confidence interval for $\lambda_{s}$.

Let $\left\{ X(t)\right\}$ be a Poisson process with intensity $\lambda$ and denote by $T_{1}$ the time of the first event. Then, $\mathbb {P}\left(T_{1} \leq s\right) = 1-\exp (-\lambda s)$. Show that, for $0 < s < t$,

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 $t$, $0 \leq t \leq 24$.

A muffin with raisins may be viewed as part of a three-dimensional Poisson process with raisins as the points. How many raisins does a muffin need to contain, on the average, in order to make the probability that a muffin contains no raisins at most 0.05? What objection can you find against the use of a Poisson process for raisins muffins?

Germ-grain models are popular as models for spatial structures, for example in ecology and material science. In a simple germ-grain model, points are distributed over a region according to a Poisson process with intensity $\lambda$. With the Poisson points as centers, one places discs (or spheres in $\mathbb {R}^{3}$) with random sizes, independently of each other. One could think of trees of random size located in a forest according to a Poisson process.

A simple question is then: what is the probability that a point chosen at random is covered by at least one disc? Find that probability if the radiuses of discs are independent and uniformly distributed between 0 and $a > 0$.

Prove that every continuous function $m(t)$ that satisfies

is of the form $m(t) = kt$.