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?未知题型
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?未知题型

类似问题

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:

Which ONE of the following thalamic nuclei is targeted by the trigeminothalamic tract?

Which ONE of the following is disinhibited in the direct pathway of the basal ganglia motor loop

30. What is the main purpose of the lecture?

Section FChoose a, b or c as the best answer.29. Which statement is the speaker most likely to agree with?

28. 15-minute neighbourhoods provide residents with inexpensive ____ .

27. ____ % of global car ownership is in the U.S.A.

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单