Question

(a) Let \(\\{N(t), t \geqslant 0\\}\) be a nonhomogeneous Poisson process with mean value function \(m(t) .\) Given \(N(t)=n\), show that the unordered set of arrival times has the same distribution as \(n\) independent and identically distributed random variables having distribution function $$ F(x)=\left\\{\begin{array}{ll} \frac{m(x)}{m(t)}, & x \leqslant t \\ 1, & x \geqslant t \end{array}\right. $$ (b) Suppose that workmen incur accidents in accordance with a nonhomogeneous Poisson process with mean value function \(m(t) .\) Suppose further that each injured man is out of work for a random amount of time having distribution F. Let \(X(t)\) be the number of workers who are out of work at time \(t\). By using part (a), find \(E[X(t)]\)

Short answer

In part (a) of the problem, it is shown that the unordered set of arrival times in a nonhomogeneous Poisson process with mean value function \(m(t)\) has the same distribution as \(n\) independent and identically distributed random variables with distribution function \(F(x) = \frac{m(x)}{m(t)}\). In part (b), the expected number of workers out of work at time \(t\), denoted by \(E[X(t)]\), is found to be \(E[X(t)] = E[F] \int_0^t \frac{\lambda(x)}{\lambda(t)} m(x) dx\), where \(E[F]\) is the expected time out of work for each injured worker, and \(\lambda(x)\) is the intensity of the nonhomogeneous Poisson process.

Step-by-step solution

Part (a)

First, let's denote the arrival times by \(T_1, T_2, \ldots, T_n\) given that \(N(t) = n\). Since the process is a nonhomogeneous Poisson process, the probability of having \(n\) arrivals until time \(t\) is given by $$ P(N(t) = n) = \frac{e^{-m(t)}m(t)^n}{n!}. $$ Now consider the joint probability density function of arrival times \(T_1, T_2, \ldots, T_n\). We can write it as $$ f(t_1, \ldots, t_n) = \frac{n!}{m(t)^n}\frac{1}{n!} \frac{\partial^n}{\partial t_1\cdots \partial t_n}P(T_1\le t_1,\ldots, T_n\le t_n). $$ Since \(N(t)\) is the number of arrivals until time \(t\), we have $$ P(T_1\le t_1,\ldots, T_n\le t_n) = P(N(t_n) = n) = \frac{e^{-m(t_n)}m(t_n)^n}{n!}. $$ Therefore, the joint probability density function can be expressed as $$ f(t_1, \ldots, t_n) = \frac{n!}{m(t)^n}\frac{\partial^n}{\partial t_1\cdots \partial t_n} \frac{e^{-m(t_n)}m(t_n)^n}{n!}. $$ Now, taking the partial derivatives, we get $$ f(t_1, \ldots, t_n) = \frac{n!}{m(t)^n} e^{-m(t_n)} \prod_{i=1}^{n} m'(t_i). $$ With this result, we can write the joint distribution function as $$ F(t_1, \ldots, t_n) = \frac{1}{m(t)^n} \prod_{i=1}^{n} m(t_i). $$ Notice that \(F(t_1, \ldots, t_n)\) defines a distribution of \(n\) independent and identically distributed random variables with distribution function $$ F(x) = \frac{m(x)}{m(t)}. $$ This completes the proof for part (a).

Part (b)

We are given that the time out of work for each injured worker follows the distribution function F. From part (a), we know that the arrival times have the same distribution. Let's denote the expected time out of work with \(E[F]\). The expected number of workers, \(E[X(t)]\), who are out of work at time \(t\) can be expressed as the sum of probabilities that a worker who arrived at time \(t_i\) is still out of work at time \(t\). Since the workers have independent arrival times, we can compute this as $$ E[X(t)] = E[F]\sum_{i=1}^{N(t)} P(T_i \le t) = E[F] \int_0^t P(T_i \le x) dN(x). $$ Given the result from part (a), we have, for \(x \leqslant t\), $$ P(T_i \le x) = \frac{m(x)}{m(t)}. $$ Therefore, we get $$ E[X(t)] = E[F] \int_0^t P(T_i \le x) dN(x) = E[F] \int_0^t \frac{m(x)}{m(t)} dN(x). $$ Now, we can use the mean value function of the nonhomogeneous Poisson process, \(N(t)\), to compute the expected number of workers out of work at time \(t\). Since \(N'(t) = \lambda(t)\), where \(\lambda(t)\) is the intensity of the process, we can write $$ E[X(t)] = E[F] \int_0^t \frac{m(x)}{m(t)} N'(x) dx = E[F] \int_0^t \frac{\lambda(x)}{\lambda(t)} m(x) dx. $$ This is the expected number of workers out of work at time \(t\), given the nonhomogeneous Poisson process with mean value function \(m(t)\).

Key concepts

Mean Value Function

The mean value function, denoted as \(m(t)\), is a key concept in the study of nonhomogeneous Poisson processes. Unlike the homogeneous Poisson process, where events occur at a constant rate over time, the nonhomogeneous Poisson process allows the rate to change. This variability is captured by the mean value function, which represents the expected number of events that occur in the interval \([0, t]\).

For a better understanding, let's liken the mean value function to a hiker's journey ascending a mountain. As the hiker moves forward in time (akin to the variable \(t\)), the elevation gained (the number of events or arrivals) might accelerate or decelerate based on the slope (the rate function of the process) at any given point. The mean value function, thus, encapsulates the summative path of the hike up to time \(t\), integrating all the fluctuations in elevation gain — or in the case of our process, the rate of events.

Mathematically, for a nonhomogeneous Poisson process, the mean value function can often be derived from the intensity function \(\lambda(t)\), which indicates the instantaneous rate of occurrence at any time \(t\). The mean value function is the integral of the intensity function over \([0, t]\), which gives:
\[m(t) = \textstyle\int_{0}^{t} \textstyle\textstylelambda(s) ds.\]
Understanding the mean value function is crucial to solving many problems related to nonhomogeneous Poisson processes, as it forms the foundation for determining distributions and probabilities of events over time.

Probability Density Function

The probability density function (PDF) is an essential element of understanding continuous probability distributions, including those that arise in nonhomogeneous Poisson processes. The PDF describes the likelihood of a random variable taking on a particular value. For each infinitesimally small interval, it provides the probability that the random variable falls within that interval.

Imagine you're sprinkling seeds over a stretch of soil. The PDF would represent the density of seeds at every point along the ground — some areas might have more seeds, others less, reflecting the probability of finding a seed in any small patch of soil.

In the realm of nonhomogeneous Poisson processes, the joint PDF of arrival times \(f(t_1, \textellipsis, t_n)\) shares the collective likelihood of \(n\) arrivals happening at specific times \(t_1, t_2, \textellipsis, t_n\). This function is crucial when we want to explore the stochastic behavior of the arrival times within the process.

Relationship to the Mean Value Function

Relating back to the mean value function, we can use it to express the joint probability density function of the arrival times. Assuming arrival times \(T_1, T_2, \textellipsis, T_n\), as conditional upon there being exactly \(n\) arrivals by time \(t\), the density function is derived by differentiating the probability with respect to each arrival time:
\[f(t_1, \textellipsis, t_n) = \textstyle\frac{n!}{m(t)^n} e^{-m(t_n)} \textstyle\prod_{i=1}^{n} m'(t_i).\]
This density function is fundamental when computing probabilities for specific patterns of arrivals within a given timeframe.

Distribution Function

The distribution function, or cumulative distribution function (CDF), is another crucial statistical concept. It quantifies the probability that a random variable is less than or equal to a specific value. Essentially, it offers a running total of probabilities, building up as you move along the random variable’s range. To visualize it, think of filling a glass with water: the level of water represents the accumulation of probability up to that point, increasing as more water (probability) is added.

Within the context of a nonhomogeneous Poisson process, the distribution function is particularly interesting. Given \(n\) arrivals at times \(T_1, T_2, \textellipsis, T_n\), by time \(t\), the CDF, \(F(x)\), defines the proportion of the mean value function up to point \(x\), relative to time \(t\):
\[F(x) = \left\{\begin{array}{ll} \frac{m(x)}{m(t)}, & x \textstyle\leqslant t \ 1, & x \textstyle\geqslant t \end{array}\right.\]
Here, the CDF \(F(x)\) describes the distribution of an individual arrival time within the process, assuming that there have been exactly \(n\) arrivals by time \(t\). This concept plays a key role in characterizing the process, as it allows us to calculate probabilities associated with the timing of events and, as seen in the textbook example, to determine expectations regarding time-dependent quantities such as the number of workers out of work at any given time.