I'm trying to solve this question starting with problem 1, but I'm not sure how to construct the joint distribution. I assume that we need to find $P[X_i < Y_i]$, but I do not know how to find this probability. Could you give me some advice for this? Also I would really appreciate if you give me advice how to approach to the rest of questions. Thank you!

John and Micheal are waiting at the bus stop outside of their dorm.

Unfortunately, the bus system is unreliable, so the length of these intervals are random, and follow Exponential

distributions.

John is waiting for the 51B, which arrives according to an Exponential distribution

with parameter $\lambda$ . That is, if we let the random variable $X_i$ correspond

to the difference between the arrival time i th and i-1st bus (also known as the inter-arrival time)

of the 51B, $X_i \sim \operatorname{Expo}(\lambda)$ .

Micheal is waiting for the 79, whose inter-arrival time, follows an Exponential distributions

with parameter $\mu$ . That is, $Y_i \sim \operatorname{Expo}(\mu)$ . Assume that all inter-arrival times are independent.

1.What is the probability that Micheal's bus arrives before John's bus?

2.After 20 minutes, the 79 arrives, and Micheal rides the bus. However, the 51B still hasn't arrived yet. Let

Let D be the additional amount of time John needs to waitfor the 51B to arrive. What is the distribution of D?

3. Lavanya isn't picky, so she will wait until either the 51B or the 79

bus arrives. Solve for the distribution of Z, the amount of time Lavanya

will wait before catching the bus.

4.Khalil arrives at the bus stop, but he doesn't feel like riding the bus with John. He

decides that he will wait for the second arrival of the 51B to ride the bus. Find the distribution

of $T = X_1 + X_2$ , the amount of time that Khalil will wait to ride the bus.

[HINT: One way to approach this problem would be to compute the CDF of T and then differentiate the CDF.]