Suppose that a company has determined that the the number of...

Suppose that a company has determined that the the number of jobs per week, $N$, varies from week to week and has a Poisson distribution with mean $\lambda$. The number of hours to complete each job, $Y_{i},$ is gamma distributed with parameters $\alpha$ and $\beta$. The total time to complete all jobs in a week is $T=\sum_{i=1}^{N} Y_{i} .$ Note that $T$ is the sum of a random number of random variables. What is a. $E(T | N=n) ?$ b. $E(T)$, the expected total time to complete all jobs?

Suppose that a company has determined that the the number of jobs per week, $N$, varies from week to week and has a Poisson distribution with mean $\lambda$. The number of hours to complete each
job, $Y_{i},$ is gamma distributed with parameters $\alpha$ and $\beta$. The total time to complete all jobs in a week is $T=\sum_{i=1}^{N} Y_{i} .$ Note that $T$ is the sum of a random number of random variables. What is
a. $E(T | N=n) ?$
b. $E(T)$, the expected total time to complete all jobs?

Key Concepts

Compound Distribution (Random Sum)

A compound distribution represents the sum of a random number of independent and identically distributed random variables. In this case, T is the compound sum where N (the random number of jobs) follows a Poisson distribution and each job duration follows a Gamma distribution. Understanding compound distributions is key to determining overall measures like mean and variance.

Poisson Distribution

The Poisson distribution models the number of events (such as jobs in this problem) occurring in a fixed interval of time when these events happen with a constant mean rate and independently of the time since the last event. Its mean and variance are both equal to the rate parameter ?, and this property is crucial for finding the expected number of jobs.

Gamma Distribution

The Gamma distribution is a two-parameter family of continuous probability distributions often used to model waiting times or durations. In this context, each job's completion time is modeled using the gamma distribution, which has an expected value given by the product of its parameters (?/? or ?? depending on the parameterization), thus allowing for the calculation of the expected job duration.

Conditional Expectation

Conditional expectation involves computing the expected value of a random variable given that another variable takes on a specific value. In this question, E(T|N=n) is calculated by conditioning on the number of jobs (N) and then summing the individual expected completion times of the jobs.

Law of Total Expectation

The law of total expectation states that the overall expectation of a random variable can be found by taking the expectation of its conditional expectation over the distribution of the conditioning variable. This law is used here to compute the overall expected total time E(T) by averaging E(T|N) over the distribution of N.

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