You are waiting for the bus, and are rather impatient. your friend tells you that busses arrive according to an exponential distribution with parameter =1/10 (busses/min) for a mean waiting time of 10 minutes. you've been waiting for a while, and want to prove their hypothesis wrong: you're thinking the wait is longer than that (one-tailed). (note: you may use the approximation e^3≈20 when applicable on this problem. you may do calculations in python if you wish, but all exposition should be in markdown.)

part a: state the null and alternative hypotheses being tested.

part b: devise a test of the form "reject if x> cx> c" where cc is how long you have to wait until the bus arrives. use a significance of α=.05α=.05. how long do you wait before you reject the null hypothesis with a probability of type i error of 5%?

part c: what is the power of your test if the true rate is λ=1/60λ=1/60? (nb: power is defined as the complement of making a type ii error: it's the rate at which you correctly reject a false null hypothesis)

part d: it turns out neither the null nor alternative are correct! the true distribution of busses was a continuous uniform distribution with pdf