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) = 0.35e−0.35x. 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 12 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 5 and 7 minutes to pay the toll? (Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.)

Answer 1

a. What is the mean waiting time that drivers face when entering San Francisco via the Golden Gate Bridge?

We are given

f(x) = 0.35*exp(-0.35*x)

Comparing it with

f(x) = λ*exp(-λ*x)

We get

λ = 0.35

Formula for mean for exponential distribution is given as below:

Mean = 1/λ

Mean = 1/0.35

Mean = 2.857143

Mean = 2.86 Minutes

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

Here, we have to find P(X>2.86)

P(X>2.86) = 1 – P(X<2.86)

P(X<2.86) = 0.632488

(By using exponential table or excel)

P(X>2.86) = 1 – P(X<2.86)

P(X>2.86) = 1 – 0.632488

P(X>2.86) = 0.367512

Required probability = 0.3675

c. What is the probability that a driver spends more than 12 minutes to pay the toll?

Here, we have to find P(X>12)

P(X>12) = 1 – P(X<12)

P(X<12) = 0.985004

(By using exponential table or excel)

P(X>12) = 1 – P(X<12)

P(X>12) = 1 – 0.985004

P(X>12) = 0.014996

Required probability = 0.0150

d. What is the probability that a driver spends between 5 and 7 minutes to pay the toll?

Here, we have to find P(5<X<7)

P(5<X<7) = P(X<7) – P(X<5)

P(X<7) = 0.913706

(By using exponential table or excel)

P(X<5) = 0.826226

(By using exponential table or excel)

P(5<X<7) = P(X<7) – P(X<5)

P(5<X<7) = 0.913706 - 0.826226

P(5<X<7) = 0.08748

Required probability = 0.0875