Question

Assume that Alice will arrive home this evening at a random time, uniformly distributed between 5pm and 6pm. Bob promises to call Alice “after 5pm”, which means Bob will wait an exponential amount of time after 5pm with expected value 30 minutes and then call Alice. Assume the time Alice arrives home is independent of the time when Bob will call.

(a) Compute the probability that Alice will not miss Bob’s call.

(b) Compute the probability that Bob will call before 6, given that Alice does not miss Bob’s call.

Answer 1

Let us denote 'bpm by 'o' and 'bpm by 'y' s - x time at which Alice will arrive home. = xn uniform (0,1). y time when Bob will make a call = you exponential (2=430) 0 Proto that Alice will not miss Bob's call = P(xcY) = P(xcy |Y) fyly) dy = ŕ P(xcy) <y) fyly) dy & 1 1. fyly) ay Hont ontol J Fx (y) fyly dy & J tylysdy Ty a ēdy dy + [1 - Fy (3)] Il tett + e a 0.98 (ALS) = 30x )+ ē 30 - 0.0005433 & Prob that Bob will call before 6 given that Alice does not mis Bols ☺ E P(X204 x<4) Caetya - PCycan XcY) PCOLXcy(1) PCOCXL P(xcY) P(XCY) P(xcy) Pay e tydy P(XLY) - Stétat P(Xcy) 30 X 0.98 0.0005433 = 0.0166 (en)