Question

When crossing the Golden Gate Bridge, traveling into San Francisco, all drivers must pay a toll. Suppose the amount of time (in minutes) drivers wait in line to pay the toll follows an exponential distribution with a probability density function of f(x) = 0.49e−0.49x.

a. What is the mean waiting time that drivers face when entering San Francisco via the Golden Gate Bridge? (Round your answer to 2 decimal places.)

b. What is the probability that a driver spends more than the average time to pay the toll? (Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.)

c. What is the probability that a driver spends more than 11 minutes to pay the toll? (Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.)

d. What is the probability that a driver spends between 6 and 10 minutes to pay the toll? (Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.)

Answer 1

Exponential distribution:

Probability density function

Mean of x :

Distribution function

Given,

X : amount of time (in minutes) drivers wait in line to pay the toll

X follows Exponential distribution with

Distribution function

a. Mean waiting time that drivers face when entering San Francisco via the Golden Gate Bridge i.e

Mean of X :  E(X) = 1/ = 1/0.49 = 2.04

Mean waiting time that drivers face when entering San Francisco via the Golden Gate Bridge = 2.04 minutes

b.probability that a driver spends more than the average time to pay the toll

From a. average time to pay the toll = 2.04

probability that a driver spends more than the average time to pay the toll = P(X>2.04) = 1-P(X2.04)

P(X>2.04) = 1-P(X2.04) = 1-0.6321 = 0.3679

probability that a driver spends more than the average time to pay the toll = 0.3679

c. probability that a driver spends more than 11 minutes to pay the toll = P(X>11) = 1-P(X11)

P(X>11) = 1-P(X11) = 1 - 0.9954 = 0.0046

probability that a driver spends more than 11 minutes to pay the toll = 0.0046

d. probability that a driver spends between 6 and 10 minutes to pay the toll = P(6X10) = P(X10) - P(X6)

P(6X10) = P(X10) - P(X6) = 0.9926-0.9471 = 0.0455

probability that a driver spends between 6 and 10 minutes to pay the toll = 0.0455