Question

I am waiting for two friends to arrive at my house. The time until A arrives is exponentially distributed with rate λa, and the time until B arrives is exponentially distributed with rate λb. Once they arrive, both will spend exponentially distributed times, with respective rates μa and μb at my home before departing. The four exponential random variables are independent.

Find the expected times of both the first and the second departures. Assume for simplicity that λa = λb = λ and µa = µb = µ.v

Answer 1

The expected time for the first departure is 1/μ and for the second departure, due to the memoryless property of the exponential distribution, it is also 1/μ.

Step-by-step explanation:

The question asks about finding the expected times of both the first and the second departures of friends A and B from a house, assuming they arrive and spend time at the house according to independent exponential distributions with identical rates for arrivals (λ) and time spent (μ). Since arrivals and departures are modeled with exponential distributions, we can use the memoryless property to address this problem.

For the first departure, both friends have the same rate of arrival and duration of stay. Therefore, the first departure is expected after the mean of the exponential distribution, which is 1/μ, since the rate of staying (departure) is identical for both friends.

For the second departure, it depends on who arrived first and the remaining time they will spend. However, due to the memoryless property of the exponential distribution, the expected remaining time for the second departure after the first departure is again 1/μ, independent of the arrival order.

Answer 2

To find the expected times of both the first and second departures, we need to calculate the expected values of the exponential random variables representing the time until A arrives, B arrives, A spends at the house, and B spends at the house.

Step-by-step explanation:

To find the expected times of both the first and second departures, we need to calculate the expected values of the exponential random variables representing the time until A arrives, B arrives, A spends at the house, and B spends at the house. The expected value of an exponentially distributed random variable with rate λ is equal to 1/λ. Therefore, the expected time of the first departure is 1/λa + 1/μa, and the expected time of the second departure is 1/λb + 1/μb.