The Department of Motor Vehicles (DMV)can service customers at a rate of 20 per hour (or 1/3 per minute)when it comes to license renewals.The service time follows an exponential distribution.What is the probability that it will take between 2 and 3 minutes to be served?
A)0.4831
B)0
C)1
D)0.1419
E)0.6284

Question

Quizplus · Accepted Answer

To solve this problem, we need to use the cumulative distribution function (CDF) of the exponential distribution which is given by:

F(x) = 1 - e^(-λx)

where λ is the rate parameter of the distribution.

In this case, the rate parameter is λ = 1/3 (since the DMV can service 1/3 customers per minute).

We want to find the probability that it takes between 2 and 3 minutes to be served. This is equivalent to finding P(2 ≤ x ≤ 3), where x is the service time.

Using the CDF, we can calculate:

P(2 ≤ x ≤ 3) = F(3) - F(2)
              = (1 - e^(-λ(3))) - (1 - e^(-λ(2)))
              = e^(-2/3) - e^(-1)

Using a calculator, we get:

P(2 ≤ x ≤ 3) ≈ 0.1419

Therefore, the answer is D) 0.1419.

The Department of Motor Vehicles (DMV)can Service Customers at a Rate