Here is a practice question that I wanted to try out for my probability course. At a first glance, I thought it would be fairly simple, but I got stuck right from the beginning. I have a few comments right after the questions. So if anyone could verify if my intuition is correct as well as help solve the problem, it would be much appreciated. Thank you for any help.
Person A has $2$ jobs that he/she will process sequentially. The first job takes an exponential amount of time with a mean of $15$ minutes. The second job also takes an exponential amount of time to complete with a mean of $15$ minutes. He/she can work on the second job only after the first job is completed.
Comment: Would it be safe to assume that the jobs are independent of each other even though we can't start the second job without first completing the first?
1.) Let $T$ be the total time to complete both jobs. Give the distribution of $T$ , its mean and its standard deviation.
Comment: I'm assuming that the distribution of $T$ would just be the sum of the two exponential distributions. Is this correct? But then I know that the exponential distribution has a memoryless property. For the mean, since they are independent jobs, would it just be $(1/μ_1)+(1/μ_2)$? Then for the variance, would it also be $\text{Var (job 1)} + \text{Var (job 2)}$ (again by independence)?
2.) Compute the probability that it would take more than $45$ minutes to complete both jobs.
Comment: I'm pretty sure that shouldn't be difficult once I've established $T$'s distribution.
3.) Suppose that person $B$ takes an exponential amount of time to process both jobs (i.e. both, not each) with a mean time of $30$ minutes. Compute the probability that it would take person B more than $45$ minutes to complete both jobs.
4.) Suppose that person A and person B start to work at the same moment. Give the probability that person B will complete both jobs before person A.
0) Yes.
1) Yes, you reasoning seems fair. Notice that, if I call the first job $X$, the second $Y$, then $$X+Y = T\sim \text{Gamma}(n = 2,\lambda = 1/15).$$
2) Sure, it asks $P(T>45)$.
3) Let $Z$ be the distribution of the time is takes person $B$ to complete the tasks. Then the distribution of $Z$ is exponential with mean $30$ min, and the questions asks $P(Z>45)$.
4) In other words, $P(Z<T)$. You could use integration, or you could use competing exponentials to try to get an answer faster.