In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. If the waiting time (in minutes) at each stop has a uniform distribution with and , then it can be shown that the total waiting time Y has the pdf A 5 0 B 5 5 a 1 2 25 , a , a 1 2 , 35 m

A 5 25 B 5 35 f(x) 5 e k[1 2 (x 2 3)2

] 2 # x # 4 0 otherwise

a. Sketch a graph of the pdf of Y.

b. Verify that .

c. What is the probability that total waiting time is at most 3 min?

d. What is the probability that total waiting time is at most 8 min?

e. What is the probability that total waiting time is between 3 and 8 min?

f. What is the probability that total waiting time is either less than 2 min or more than 6 min?