Customers arrive at a retail store at an average rate of five per hour.The time between arrivals can be described by an Exponential distribution.Assuming a customer has just arrived (time = 12:00 pm),find the arrival time for next two customers.Using the following random digits (left to right): 27,05
A)12:16 PM.and 12:36 PM
B)12:27 PM and 12:32 PM
C)12:05 PM and 12:27 PM
D)12:32 PM and 12:59 PM

Question

Quizplus · Accepted Answer

The time between arrivals follows an Exponential distribution with a rate parameter of λ=5 customers per hour. We can use the formula for the Exponential distribution to find the probability density function:

f(x) = λe^(-λx)

where x is the time between arrivals. We can use this probability density function to find the cumulative distribution function (CDF):

F(x) = 1 - e^(-λx)

The CDF gives us the probability that the time between arrivals is less than or equal to x. We can use this to generate random arrival times using the given random digits:

First customer: Time = 12:00 pm (given)
Second customer: 
Using random digits 27, we have:
U = 0.27 (reading digits in pairs and dividing by 100)
F(x) = 1 - e^(-5x)
0.27 = 1 - e^(-5x)
e^(-5x) = 0.73
-5x = ln(0.73)
x ≈ 0.16 (in hours)
So, the arrival time of the second customer is 12:00 pm + 0.16 hours = 12:16 pm.

Third customer: 
Using random digits 05, we have:
U = 0.05
F(x) = 1 - e^(-5x)
0.05 = 1 - e^(-5x)
e^(-5x) = 0.95
-5x = ln(0.95)
x ≈ 0.36 (in hours)
So, the arrival time of the third customer is 12:16 pm + 0.36 hours = 12:36 pm.

Therefore, the answer is A)12:16 PM and 12:36 PM

Customers Arrive at a Retail Store at an Average Rate