Arrival times given the count as uniform order statistics

Event $A$ occurs $m$ times during a time period $T_{0}$, with the probability of its occurrence being the same at every moment. Determine the probability that it occurred $k$ times during a given interval $T$, letting $T_{0} \to \infty$ and $\frac{m}{T_{0}} \to p$.

This exercise asks you to prove that the Poisson process conditioned to be equal to $n$ at time 1 has the same law as $n$ times the empirical process. Here is the precise statement. Suppose $P_{t}$ is a Poisson process with parameter $\lambda >0$. Let $\mathbb {Q}$ be the law of $\left\{ P_{t}, t \in [0,1]\right\}$ conditioned so that $P_{1}=n$. Thus $\mathbb {Q}$ is a probability on $D[0,1]$ with

Since $\left(P_{1}=n\right)$ is an event with positive probability, there is no difficulty defining these conditional probabilities. Prove that $\mathbb {Q}$ is also the law of $\left\{ n F_{n}(t), t \in [0,1]\right\}$, where $F_{n}$ is defined in Section 35.3.

For the Poisson process (see Example 33.1) show that for $0<s<t$,

Thus the conditional distribution (in the sense of Theorem 33.3) of $N_{s}$ given $N_{t}$ is binomial with parameters $N_{t}$ and $s / t$.

$33.734 .11 \uparrow$ Use Example 33.10 to calculate $P\left[N_{s}=k \| N_{u}, u \geq t\right](s \leq t)$ for the Poisson process.

Let $X$ follow a Poisson distribution with parameter $\lambda$. We select $N$ random points from the interval $[0, 1]$. Let $X_{i}$ be the number of points in the interval $[(i-1)/n, i/n]$ ($i = 1, \ldots , n$). Show that the random variables $X_{i}$ are independent and identically distributed.

Equation (2.42) gives $f_{S_i \mid N(t)}(s_i \mid n)$, which is the density of rv $S_i$ conditional on $N(t) = n$ for $n \geq i$. Multiply this expression by $\mathbb {P}\left(N(t) = n\right)$ and sum over $n$ to find $f_{S_i}(s_i)$; verify that your answer is indeed the Erlang density.

Let $S_{m} = \sum_{r = 1}^{m} X_{r}, m \leq n$, where the $X_{r}$ are independent and identically distributed with finite mean. Denote by $U_{1}, U_{2}, \ldots , U_{n}$ the order statistics of $n$ independent variables which are uniformly distributed on $(0, t)$, and set $U_{n+1} = t$. Show that $R_{m} = S_{m} / U_{m+1}, 0 \leq m \leq n$, is a backward martingale with respect to a suitable sequence of $\sigma$-fields, and deduce that

Let $\left\{ X_{r}: r \geq 1\right\}$ be independent exponential random variables with parameter $\lambda$, and set $S_{n} = \sum_{r = 1}^{n} X_{r}$. Show that:

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

Suppose that we are at time zero. Passengers arrive at a train station according to a Poisson process with intensity $\lambda$. Compute the expected value of the total waiting time of all passengers who have come to the station in order to catch a train that leaves at time $t$.

Suppose that electrical pulses having i.i.d. random amplitudes $A_1, A_2, \ldots$ arrive at a counter in accordance with a Poisson process with intensity $\lambda$. The amplitude of a pulse is assumed to decrease exponentially, that is, if a pulse has amplitude $A$ upon its arrival, then its amplitude at time $t$ is $Ae^{-\alpha t}$, where $\alpha$ is some positive parameter. We finally assume that the initial amplitudes of the pulses are independent of the Poisson process. Compute the expected value of the total amplitude at time $t$.

Let $\left\{ X(t), t \geq 0\right\}$ be a Poisson process and let $\left\{ T_k, k \geq 1\right\}$ be the occurrence times. Suppose that we know that $T_3=1$ and that $T_1=x$, where $0 < x < 1$. Our intuition then tells us that the conditional distribution of $T_2$ should be $\mathrm{U}(x,1)$-distributed. Prove that this is indeed the case, i.e., show that

A train departs at time $s > 0$. Starting from time $t=0$, passengers arrive in a Poisson stream with intensity $\lambda$. Prove that the mean total waiting time of the passengers equals $\lambda s^{2}/2$.

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 $(1,2]$.

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 $0 < s < t$,

Denote by $v_{t}$ the random number of requests of a Poisson flow that arrive in the interval $[0, t)$. Prove that for $t \leqslant T, k \leqslant n$,

Prove that if it is known that $N$ requests of a Poisson flow arrived on the segment $[0, T], T > 0$, then the flow of requests on this segment is a Bernoulli flow.