100 students are waiting for service at the Office of Enrollment. The waiting ti

Question

100 students are waiting for service at the Office of Enrollment. The waiting time T, in minutes, of an individual student has an exponential distribution with parameter =3 and the waiting times for different students are independent. It can be shown that the total waiting time, X, of the 100 students (in minutes) has a gamma distribution with shape parameter 100 and scale parameter 1/3. (In R this is shape=100, scale = 1/3. In the text this corresponds to = 100, =1/3)

a) What is the expected value of X in minutes?

b) What is the variance of X?

c) What is the 80th percentile of X?

d) What is the probability that 33 X 38?

e) What is the probability that the total waiting time is more than 40 minutes?

f) What is the probability that the total waiting time is at most 30 minutes?

g) What is the probability that X is within 1 standard deviation of its expected value?

Explanation / Answer

here it is a gamma distribution

hence )

a) mean = =100/3

b) variance =2 =100/9

c) 80th percentile of X =36.1015

d) P(33<X<38) =0.9150-0.4733 =0.4417

e)P(X>40) =1-P(X<40)=1-0.9721 =0.0279

f)P(X<30)=0.1582

g) P(100/3-10/3 <X<100/3+10/3) =P(30<X<36.6667)=0.8417-0.1582 =0.6835

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.