Question

The time between calls to a corporate office is exponentiallydistributed with a mean of 10 minutes. Find:

a.) fx(X)

b.) Probability that there are no calls within one-half hour?

c.) Given that you have already been waiting for half an hour. How long do you expect to wait until the next call?

Answer 1

here expected mean arrival time = =10

a)f(X)=(1/)*e^-x/     for x>0

b) Probability that there are no calls within one-half hour (30 minutes)=P(X>30)=e^-30/10 =e^-3

^=0.0498

^c)as exponential distribution is memoryless:

expect to wait until the next call =10 minutes

Answer 2

To solve the given questions, we'll use the exponential distribution with a mean of 10 minutes. The exponential distribution has the probability density function (PDF) given by:

f(x) = λ * e^(-λx)

where λ is the rate parameter, which is the reciprocal of the mean. In this case, λ = 1/10 = 0.1.

a) To find the probability density function (PDF) of the exponential distribution, fx(X), we substitute the values into the formula:

f(x) = 0.1 * e^(-0.1x)

b) The probability that there are no calls within one-half hour (30 minutes) can be calculated by finding the cumulative distribution function (CDF) up to 30 minutes:

P(X > 30) = 1 - P(X ≤ 30)

Using the exponential distribution's CDF formula:

F(x) = 1 - e^(-λx)

P(X > 30) = 1 - F(30) = 1 - (1 - e^(-0.1 * 30)) = e^(-0.1 * 30)

c) Given that you have already been waiting for half an hour (30 minutes), the waiting time until the next call follows the exponential distribution with the same mean of 10 minutes. Therefore, the expected waiting time can be calculated as the mean of the exponential distribution, which is 10 minutes.

So, if you have already been waiting for half an hour, you can expect to wait for another 10 minutes until the next call.