Question

The counters at a city hall are being modernized. The city hall has 20 counters. Four
workmen have been called in to perform the conversion. Each workman converts 5
counters. During the work, a counter is closed for an average of 3 hours. The duration of
the closing of a counter can be approached using an exponential distribution. What is the
probability for each workman that he/she will be finished with the work within one 8-hour
workday?

Answer 1

A worker has to finish modernizing 5 counters.

Let Xi be the time taken to modernize the counter i by a workman. i=1,2,3,4,5.

It is given that,

, iid (since, each counter are independant)

Suppose X1, X2, ... , Xn are n mutually independent random variables having exponential distribution with parameter .
Define

Then, the sum Z is a Gamma random variable with parameters n and .

Thus,

The pdf of Z is given by:
  
Now, we are to calculate,


The integral is very complex to calculate. However, they can be calculated using Minitab 17. The steps are shown below:

After opening Minitab, we select Calc then select Probability Distributions and then select Gamma...
A dialogue box opens named Gamma Distribution.
We select Cumulative Probability.
We set the Shape Parameter to 5.
We set the Scale Parameter to 3.
We set the Threshold Parameter to 0.0.
We set Input Constant as 8 and click on OK.
We get the result in Session window.

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