Question

Ex. 64

Suppose your waiting time for a bus in the morning is uniformly distributed on [0, 8], whereas waiting time in the evening is uniformly distributed on [0, 10] independent of morning waiting time.

a. If you take the bus each morning and evening for a week, what is your total expected waiting time? [Hint: Define rv's ?1,…,?10 and use a rule of expected value.]

b. What is the variance of your total waiting time?

c. What are the expected value and variance of the difference between morning and evening waiting times on a given day?

d. What are the expected value and variance of the difference between total morning waiting time and total evening waiting time for a particular week?

Answer 1

for uniform distribution:

here mean waiting time in the morning X=(lower end point+upper end point)/2=(0+8)/2=4

here mean waiting time in the evening Y=(lower end point+upper end point)/2=(0+10)/2=5

therefore mean waiting time for the day E(T) =E(X+Y) =E(X)+E(Y) =4+5 =9 minutes