5. Finally assume that busses arrive with non-constant rate function (t) per minute, where t > 0 denotes the time measured in minutes. (a) If (t) = 1 t , find the probability that the first bus arrives within 1 minute. (b) If (t) = t, find the expected arrival time of the first bus.

This problem says, basically, that if one begins waiting at a bus stop, and busses arrive at a non-constant rate of 1/t per minute, what is the probability that one (or more) bus(ses) arrive within 1 minute.

For Part A, I tried to find the probability distribution function and probability density function by integrating from 1/t from t to 0. This integral diverges. My question is: is it better to integrate with different parameters, such as t to 1? Also, once I am able to find the probability density function, how am I supposed to apply it to this problem. I thought about using a exponential random variable, which generally uses a constant rate function i.e. GAMMA= some #, but such a technique does not return a meaningful probability?