/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 80 Let \(T_{1}, T_{2}, \ldots\) den... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 80

Let \(T_{1}, T_{2}, \ldots\) denote the interarrival times of events of a nonhomogeneous Poisson process having intensity function \(\lambda(t)\). (a) Are the \(T_{i}\) independent? (b) Are the \(T_{t}\) identically distributed? (c) Find the distribution of \(T_{1}\).

Short Answer

Expert verified
(a) The interarrival times \(T_{1}, T_{2}, \ldots\) are not independent. (b) The interarrival times \(T_{t}\) are not identically distributed. (c) The distribution of the first interarrival time, \(T_{1}\), is given by: \[f_{T_{1}}(t) = \lambda(t) e^{-\int_{0}^{t} \lambda(u) du}\]

Step by step solution

01

(a) Assess independence of \(T_i\)'s

: The interarrival times \(T_1, T_2, \ldots\) represent the time gaps between successive events in a nonhomogeneous Poisson process. To determine if these interarrival times are independent, let's recall the true definition of independence: Events are independent if the occurrence of one event has no influence on the occurrence of another event. For a nonhomogeneous Poisson process, the intervals between the arrival times depend on the intensity function, 位(t). Since 位(t) varies over time, the probability of events occurring in distinct time intervals is dependent on the time-varying intensity function. As a result, the interarrival times \(T_{1}, T_{2}, \ldots\) are not independent.
02

(b) Assess identical distribution of \(T_t\)'s

: Let's now examine if the interarrival times \(T_{t}\) have identical distributions. We know that the distribution of the interarrival times is dependent on the intensity function of the nonhomogeneous Poisson process. Since the intensity function, 位(t), varies over time, it implies that the distribution of interarrival times can also vary, depending on the time at which events occur. To be identically distributed, the interarrival times should have the same distribution, regardless of when they occur. In this case, since the distribution of interarrival times is determined by a time-varying intensity function, the interarrival times \(T_{t}\) are not identically distributed.
03

(c) Finding the distribution of \(T_1\)

: To find the distribution of the first interarrival time, \(T_{1}\), we need to use the intensity function, 位(t). We know that the cumulative distribution function (CDF) of interarrival times for nonhomogeneous Poisson processes can be found by integrating the intensity function over time: \[F_T(t) = P(T_{1} \le t) = 1 - e^{-\int_{0}^{t} \lambda(u) du}\] To find the probability density function (PDF) of the interarrival times, we take the derivative of the CDF with respect to time t: \[f_T(t) = \frac{d}{dt} F_T(t) = \frac{d}{dt} \left( 1 - e^{-\int_{0}^{t} \lambda(u) du} \right)\] Using the chain rule, we get: \[f_T(t) = \lambda(t) e^{-\int_{0}^{t} \lambda(u) du}\] Thus, the distribution of the first interarrival time, \(T_{1}\), is: \[f_{T_{1}}(t) = \lambda(t) e^{-\int_{0}^{t} \lambda(u) du}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interarrival Times
Understanding the nature of interarrival times is crucial when dealing with stochastic processes like Poisson processes. Interarrival times, denoted as T1, T2, \(\ldots\), are the times between consecutive events in a process.

In the context of a nonhomogeneous Poisson process, the interarrival times signify the periods between successive random events that do not happen with a constant average rate. Contrary to homogeneous Poisson processes, where events occur with a steady rate over time and interarrival times follow an exponential distribution and are independent, nonhomogeneous Poisson processes have a variable rate of occurrence, represented by an intensity function \(\lambda(t)\), which affects these times.

Thus, the interarrival times in a nonhomogeneous Poisson process are neither independent nor identically distributed, as they are influenced by the time-varying intensity function. Their nature challenges the intuition that may come from observing more straightforward, time-homogeneous versions of the process.
Intensity Function
The intensity function, denoted by \(\lambda(t)\), is a fundamental concept in characterizing a nonhomogeneous Poisson process. It describes the instantaneous rate at which events occur as a function of time. To visualize this, imagine the intensity function as a way to gauge how busy or quiet a process is at various moments.

For instance, consider a call center receiving phone calls. During peak hours, the intensity function would be higher, reflecting a higher rate of incoming calls. Unlike the constant intensity in a homogeneous Poisson process, \(\lambda(t)\) in a nonhomogeneous process changes over time, capturing fluctuations in activity. Consequently, this time-variant intensity regulates the probability of event occurrences within different time frames, which profoundly influences the distribution of interarrival times between these events.

This variable intensity is why interarrival times in such processes are neither independent nor identically distributed - the conditions and rate of event occurrences change and depend on when you're observing the process.
Distribution of Interarrival Times
When it comes to figuring out the distribution of interarrival times in a nonhomogeneous Poisson process, things get a bit intricate due to the variable intensity function. The distribution reflects the probability of time lengths between successive events, which in nonhomogeneous scenarios, varies over time.

To find the distribution, one starts with the cumulative distribution function (CDF) which, for the first interarrival time \(T_{1}\), is given by the integral of the intensity function up to time t - reflecting the accumulated rate at which events have occurred. Mathematically, it's expressed as:
\[F_{T}(t) = P(T_{1} \le t) = 1 - e^{-\int_{0}^{t} \lambda(u) du}\]
The probability density function (PDF), which gives a detailed picture of how probable different interarrival times are, is the derivative of the CDF.
\[f_{T}(t) = \lambda(t) e^{-\int_{0}^{t} \lambda(u) du}\]

Understanding this distribution helps predict the likelihood of the time until the next event in a process that is irregular, one that does not follow a neat pattern, and hence tailors our expectations to a more realistic model of the process at hand.

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Most popular questions from this chapter

A two-dimensional Poisson process is a process of randomly occurring events in the plane such that (i) for any region of area \(A\) the number of events in that region has a Poisson distribution with mean \(\lambda A\) and (ii) the number of events in nonoverlapping regions are independent. For such a process, consider an arbitrary point in the plane and let \(X\) denote its distance from its nearest event (where distance is measured in the usual Euclidean manner). Show that (a) \(P\\{X>t\\}=e^{-\lambda \pi t^{2}}\) (b) \(E[X]=\frac{1}{2 \sqrt{\lambda}}\)

Let \(X\) and \(Y\) be independent exponential random variables with respective rates \(\lambda\) and \(\mu\). Let \(M=\min (X, Y)\). Find (a) \(E[M X \mid M=X]\) (b) \(E[M X \mid M=Y]\) (c) \(\operatorname{Cov}(X, M)\)

Consider a single server queueing system where customers arrive according to a Poisson process with rate \(\lambda\), service times are exponential with rate \(\mu\), and customers are served in the order of their arrival. Suppose that a customer arrives and finds \(n-1\) others in the system. Let \(X\) denote the number in the system at the moment that customer departs. Find the probability mass function of \(X\). Hint: Relate this to a negative binomial random variable.

An event independently occurs on each day with probability \(p\). Let \(N(n)\) denote the total number of events that occur on the first \(n\) days, and let \(T_{r}\) denote the day on which the \(r\) th event occurs. (a) What is the distribution of \(N(n) ?\) (b) What is the distribution of \(T_{1} ?\) (c) What is the distribution of \(T_{r} ?\) (d) Given that \(N(n)=r\), show that the unordered set of \(r\) days on which events occurred has the same distribution as a random selection (without replacement) of \(r\) of the values \(1,2, \ldots, n\).

For the infinite server queue with Poisson arrivals and general service distribution \(G\), find the probability that (a) the first customer to arrive is also the first to depart. Let \(S(t)\) equal the sum of the remainihg service times of all customers in the system at time \(t\). (b) Argue that \(S(t)\) is a compound Poisson random variable. (c) Find \(E[S(t)]\). (d) Find \(\operatorname{Var}(S(t))\).
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