Question

Suppose that the length of long distance phone calls, measured in minutes, is known to have an exponential distribution with the average length of a call equal to eight minutes.

a. What is the probability density function (PDF) for the length of long distance phone calls?

b. Calculate the mean (μ) and standard deviation (σ) for the length distribution.

c. If X represents the length of a phone call, what is the probability P(X > 10)?

d. Why is the exponential distribution often used to model waiting times?

Answer 1

The probability density function (PDF) for the length of long distance phone calls is (1/8)e^(-x/8). The mean and standard deviation for the length distribution are both 8 minutes. The probability that a phone call lasts more than 10 minutes can be calculated using the exponential distribution. The exponential distribution is often used to model waiting times because it is memoryless.

Step-by-step explanation:

a. The probability density function (PDF) for the length of long distance phone calls, X, is given by f(x) = (1/8)e^(-x/8), where x >= 0. This is the exponential distribution function.

b. The average (mean) of the exponential distribution is equal to the parameter of the distribution, which is 8 minutes. The standard deviation of an exponential distribution is equal to the mean.

c. To find P(X > 10), we need to calculate the area under the PDF curve to the right of x = 10. This can be done using integration or by using the cumulative distribution function (CDF). The CDF for the exponential distribution is given by F(x) = 1 - e^(-x/8), where x >= 0. P(X > 10) = 1 - F(10).

d. The exponential distribution is often used to model waiting times because it is memoryless. This means that the probability of waiting a certain amount of time does not depend on how long you have already waited. It is useful in modeling situations where events occur randomly and independently over time.