In Hyde Park There Are Six Taxi Companies. When You Call Company #1, The Amount Of Time (In Minutes) That It Takes The Cab To Arrive Is Distributed As Exponential(1); If You Call Company #2, Then It's Distributed As Exponential(2), ..., If You Call Company #6, It's Generated As Exponential(6). (Recall That An Exponential(A) Distribution, Whose Parameter Is

In Hyde Park there are six taxi companies. When you call company #1, the amount of time (in minutes) that it takes the cab to arrive is distributed as Exponential(1); if you call company #2, then it's distributed as Exponential()..., if you call company #6, it's generated as Exponential(6). (Recall that an Exponential(A) distribution, whose parameter is the rate ), has mean 1/A and variance 1/2.) A statistician rolls a single standard die (sides numbered 1,2,3,4,5,6), and looks at the number rolled (let's call it X). Next, she calls the company chosen by the roll of the die. Let T be the amount of time that she waits for a cab to arrive. You are waiting for the cab with her, but you don't know what was the roll of the die. Write down a hierarchical model. (b) Calculate P(T > t) where t is a fixed value. (c) Calculate E(T). (d) Suppose that after t minute the cab has not yet arrived. Given this information, what is the conditional distribution of X? (e) As t goes to infinity, describe how the conditional distribution of X changes. Which taxi companies become more/less likely? (Hint: you may want to calculate explicit probabilities at a few values of t-but this is not necessary.)