Suppose that people arrive at a bus stop in accordance with...

Suppose that people arrive at a bus stop in accordance with a Poisson process with rate $\lambda .$ The bus departs at time $t .$ Let $X$ denote the total amount of waiting time of all those who get on the bus at time $t$. We want to determine $\operatorname{Var}(X) .$ Let $N(t)$ denote the number of arrivals by time $t$. (a) What is $E[X \mid N(t)] ?$ (b) Argue that $\operatorname{Var}[X \mid N(t)]=N(t) t^{2} / 12$. (c) What is $\operatorname{Var}(X)$ ?

Suppose that people arrive at a bus stop in accordance with a Poisson process with rate $\lambda .$ The bus departs at time $t .$ Let $X$ denote the total amount of waiting time of all those who get on the bus at time $t$. We want to determine $\operatorname{Var}(X) .$ Let $N(t)$ denote the number of arrivals by time $t$.
(a) What is $E[X \mid N(t)] ?$
(b) Argue that $\operatorname{Var}[X \mid N(t)]=N(t) t^{2} / 12$.
(c) What is $\operatorname{Var}(X)$ ?

Key Concepts

Poisson Process

A Poisson process is a stochastic process used to model random events occurring independently over time, with a constant average rate. It is characterized by the fact that the number of events in disjoint intervals is independent, leading to the popular Poisson distribution for counts in a fixed time interval.

Conditioning on the Number of Events

In many problems involving Poisson processes, it is useful to condition on the total number of events occurring in a given time interval. This isolates the randomness in the timing of events from the count, making it easier to compute conditional expectations and variances by treating the event times as a set of independent random variables given the count.

Uniform Distribution and Order Statistics

Given the number of events in a Poisson process over a fixed interval, the event times are distributed as the order statistics of independent uniform random variables. This fact simplifies the analysis of waiting times or other time-dependent measures, since the uniform distribution has well-known moments that can be exploited for calculations.

Conditional Expectation and Variance

Conditional expectation and variance are techniques in probability that help in analyzing random variables by first conditioning on another variable. Once the conditional moments are computed given the number of events, they can be combined with the distribution of the conditioning variable to find overall moments.

Law of Total Variance

The law of total variance is a fundamental result in probability that decomposes the variance of a random variable into the expected conditional variance plus the variance of the conditional expectation. This technique is particularly useful in problems where a random variable's behavior is first analyzed given another random variable, such as the number of events in a Poisson process.

Transcript

Please give Ace some feedback