On average, a printer takes 4 mins to print a file. You send a job to the printe

Question

On average, a printer takes 4 mins to print a file. You send a job to the printer at 10:00am, and it appears to be the 3rd in line. Using this information answer the following:

(a) Suppose you have to leave for class by 10:15am. What is the probability that you will be able to get the print out to class?

(b) What is the probability that it will take between 7 and 13 mins for your file to finish printing?

(c) Suppose you check the printer at 10:07am and your files have not been printed yet. What is the probability that they will have been printed by the time you need to leave for class?

(d) Without using a calculator or internet sites to compute the integral, compute the 5th moment of the time till your files have been printed.

(e) When are your files expected to finish printing? (f) What is the standard deviation of the time it will take for the files to finish printing?

Explanation / Answer

Let X = number of files printed by a printer in one hour.

Then, we assume, X ~ Poisson (15) [an average 4 minutes for one print => 15 prints per hour.]

Back-up Theory

If a random variable X ~ Poisson(), i.e., X has Poisson Distribution with mean then

probability mass function (pmf) of X is given by P(X = x) = e – .x/(x!) …………..(1)

where x = 0, 1, 2, ……. ,

Values of p(x) for various values of and x can be obtained by using Excel Function.

If X = number of times an event occurs during period t, Y = number of times the same event occurs during period kt, and X ~ Poisson(), then Y ~ Poisson (k) …………….. (2)

Part (a)

Given a job is sent to the printer at 10:00am, which appears to be the 3rd in line, and that the student has to leave for class by 10:15am, the probability that the student will be able to get the print out to class = P(4 print out happens in 15 minutes)

= P(Y = 4), where Y = number of files printed by the printer in 15 minutes, which vide (2) of Back-up Theory, has a Poisson (15/4).

Thus, required probability = P(Y = 4/ = 3.75) = 0.1938 ANSWER [by Excel Function of Poisson Distribution]

Part (b)

Probability that it will take between 7 and 13 minutes for the file to finish printing

= Probability that it will take 7 minutes to print the 3 waiting files and then it will take 6 minutes to print the fourth file

= Probability of 3 prints in 7 minutes and then 1 print in 6 minutes

= Probability of 3 prints in 7 minutes x probability of 1 print in 6 minutes

= P(Z = 3) x P(S = 1), where Z = number of prints in 7 minutes and S = number of prints in 6 minutes.

By (2) of Back-up Theory, Z ~ Poisson (7/4) and S ~ Poisson (6/4) and hence the answer is:

0.1552 x 0.3347 = 0.0519 ANSWER

Part (c)

Assuming that all the 3 waiting files have been printed by 10:07, the required probability

= P(X = 1 in 8 minutes)

= P(X = 1/ = 8/4 = 2)

= 0.2707 ANSWER/

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