Suppose that Xt is a Poisson process with X1 = 3 and X2 = 10; let Tn = inf{t > 0 : Xt = n}. What distribution is best suited to model the random variable T10 - T3?Unknown Question Type
A
Exponential
B
Normal
C
Brownian Motion
D
Multinomial
E
Poisson
F
Gamma

Question

Suppose that Xt is a Poisson process with X1 = 3 and X2 = 10; let Tn = inf{t > 0 : Xt = n}. What distribution is best suited to model the random variable T10 - T3?

BlackTom AI · Accepted Answer

To understand T10 − T3, think about the Poisson process as a sequence of interarrival times between successive events.
Option by option analysis:
Exponential: An exponential time represents the waiting time for a single event (the first arrival). However, T10 − T3 is the waiting time for 7 more events (from the 3rd to the 10th). Thus it is not just a single exponential, but the sum of seven exponentials, so this option is incomplete.
Normal: The normal distribution is not the natural model for waiting times in a Poisson process, especially for sums of independent exponential interarrival times. The gamma/Erlang form arises from the sum of exponentials, not from a symmetric bell curve.
Brownian Motion: Brownian motion is a continuous-time stochastic process, not a distribution for a finite positive waiting time. It describes paths, not a particular waiting-time random variable, so this is not appropriate for T10 − T3.
Multinomial: Multinomial distributions govern counts across categories in a fixed number of trials, not positive continuous waiting times between events in a Poisson process. It does not model waiting times.
Poisson: The Poisson distribution models counts in a fixed interval, not the waiting time between events. While related, it does not describe the distribution of a positive waiting time directly, so this is not the best model for T10 − T3.
Gamma: In a Poisson process with rate λ, the interarrival times are i.i.d. Exponential(λ). The time to observe 7 more events (from T3 to T10) is the sum of 7 independent Exponential(λ) variables, which follows an Erlang/Gamma distribution with shape parameter 7 and rate λ (or scale 1/λ). This is precisely the distribution that models T10 − T3.

Suppose that Xt is a Poisson process with X1 = 3 and X2 = 10; let Tn = inf{t > 0 : Xt = n}. What distribution is best suited to model the random variable T10 - T3?Unknown Question Type

Similar Questions

The Gamma distribution parameters, alpha and beta, are calculated from the mean and standard deviation of wet day data?

The Gamma distribution has two parameters, which are:

Question at position 18 Establishing a wholly-owned subsidiary provides the quickest access to a new market.TrueFalse

Question at position 17 Leading companies should recognize that a firm based in a country with a national competitive advantage is not guaranteed ............................ as it implements its chosen international business-level strategy.dominanceplansuccessNone of the Above

Question at position 16 Costs associated with the issues firms face when entering foreign markets, including unfamiliar operating environments, economic, administrative, and cultural differences, are known as regionalization costs.TrueFalse

Question at position 14 Which of the following is NOT an incentive for firms to become multinational?To gain access to consumers in emerging marketsTo gain easier access to raw materialsTo avoid high domestic taxation on corporate incomeOpportunities to integrate operations on a global scale

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler