Question

Minwoo usually suffers from his dental pain. So, he used to go to two dental clinics ‘Suduki DDS’ and ‘Student Health Dental Care Center’. After getting the ‘Root Canal’ treatments in the clinics many times, Minwoo finds out that the wait time to get the treatment in ‘Suduki DDS’ can be modeled by an exponential random variable with mean time 2 hours and the wait time to get the treatment in ‘Student Health Dental Care Center’ can be modeled by a uniform random variable from 0 to 2 hours. Assume the wait time to get the treatment of these two clinics are independent. Let the wait time to get the treatment in ‘Suduki DDS’ and ‘Student Health Dental Care Center’ be X and Y . Given X ă 2Y , find the conditional probability that 2X ą Y

Answer 1

& And let X & Y be waiting time to get treatment in 'Suduki Dos' and 'Student Health Deutul care Center' respectively Griven that X is Exp (W) and yu (0,2) Then pdf of X and Y fx(x)= dedx ; x 30 fyly) = 1/2 ; 02-422 1 = 1/2 Now since y ly are independent. Thes 24 PlX<29)= lo lo fx(x) dx ) fylgdy 1² 15 de xx dx ) I dy Idy 1/2 shtetx 12 ay = ½ l l dy-lo e I dy ] = 1 [1418t le 4 1 2] a = [2+ e- 1] = 'te = 0.5677. 1 Y fo- =

24 24 Hence the conditional probability in P(2x >y1x22y) P12X>Y, X<21) P(X<24) where PC 2XY, X<24)=zy P (<X2, 27, 02722) fxlx)dx) fyly) dy de med xj dy = 6²1-bely indy = { $? (0914.8") dy = { [ +479141720p] +ē? ] = [3+ (3+é 4 ė'12] E X0.709212644 = 0.3546 P(2x>4 X22Y) = - 1 2 1 2 l - 1/2 4 -4 e 1 Hence 0.3546 0.5677 = 0.6246