Question

The L. I. Bean catalog department that receives the majority of its orders by telephone conducted a study to determine how long customers were willing to wait on hold before ordering a product. The length of time was found to be a random variable best approximated by an exponential distribution with a mean equal to 3 minutes. 

Required

 a. What is the value of a, the parameter of the exponential distribution in this situation? What is meaning of λ?

 b. What proportion of customers having to hold more than 1.5 minutes will hang up before placing an order? 

 c. Find the waiting time at which only 10% of the customers will continue to hold. 

 d. What is the probability that a randomly selected caller is placed on hold for 3 to 6 minutes?  

Answer 1

Given that mean of the exponential distribution is 3 minutes.

a) For an exponential distribution, the probability distribution function is

Therefore, here the decay parameter is   

indicates how quicly the decay occurs.

b) to find : P(X > 1.5)

i.e 39.3 % people are willing to wait more for than 1.5 minutes. Hence 60.7 percent people will hang up before placing an order after 1.5 minutes.

c) To find :

i.e the time t0 upto which 90 percent of the customers are willing to wait.

d) to find : P( 3 < X < 6)