Question

The length of time (in minutes) spent waiting in line at a certain bank is modeled by the exponential probability density function

f(t) = { ce-ct if t ≥ 0
0 if t < 0
where c is a constant.The mean wait time is 8 minutes.
(a) What is c?
(b) What is the probability that a customer will have to wait more than 10 minutes?

Answer 1

To find the constant c for the exponential probability density function with a mean wait time of 8 minutes, we use the formula 1/c = mean, resulting in c = 1/8. The probability of waiting more than 10 minutes is found using the CDF of the exponential distribution, resulting in P(T > 10) = e^(-10/8).

Step-by-step explanation:

The student's question pertains to finding the constant c in the exponential probability density function for the wait time at a bank, and then using it to calculate the probability of a customer waiting more than 10 minutes. Given that the mean wait time is 8 minutes, the value of c can be determined using the relationship between the mean and the rate parameter in an exponential distribution, which is mean = 1/c. Therefore, c = 1/8.

To find the probability that a customer will have to wait more than 10 minutes, we use the cumulative distribution function (CDF) of the exponential distribution, which, for a wait time t, is given by 1 - e-ct. The probability of waiting more than 10 minutes is thus P(T > 10) = 1 - P(T ≤ 10) = 1 - (1 - e-(1/8)×10) = e-10/8.