There are three jobs and a single worker who works first on...

There are three jobs and a single worker who works first on job 1 , then on job 2 , and finally on job 3. The amounts of time that he spends on each job are independent exponential random variables with mean 1. Let $C_{i}$ be the time at which job $i$ is completed, $i=1,2,3$, and let $X=\sum_{i=1}^{3} C_{i}$ be the sum of these completion times. Find (a) $E[X]$, (b) $\operatorname{Var}(X)$.

Key Concepts

Exponential Distribution

This concept refers to a probability distribution that is often used to model the time between events in a Poisson process. In the context of the problem, each job’s duration is an independent exponential random variable with a mean that determines its rate parameter. The exponential distribution is characterized by its memoryless property, meaning that future probabilities are independent of past events, and it has both its mean and variance defined simply in terms of its rate parameter.

Linearity of Expectation

This principle states that the expected value of a sum of random variables is equal to the sum of the expected values of those random variables, regardless of whether they are independent. In the given problem, the sum of completion times is expressed as a linear combination of independent exponential random variables, allowing the expected total time to be computed by summing the individual expectations weighted by their respective coefficients.

Variance of Linear Combinations of Independent Random Variables

When dealing with independent random variables, the variance of a linear combination is the sum of the variances of the individual variables each multiplied by the square of their coefficients. In the problem, because the job durations are independent and each has a known variance, this property allows for the direct computation of the variance of the sum of the completion times by considering how many times each job duration appears in the overall sum.

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