Question 4) (25 points) On the third floor of the College of Engineering Building at the University of Western Arabia there is a printer used by the faculty of industrial engineering (IE) and electrical engineering (EE) departments. Print jobs are sent to the printer according to a Poisson distribution at a rate of 3 jobs per 5 minutes. An analysis of jobs that have been sent to the printer in the last year indicates that on average 60% of the jobs are from the EE department whereas the remaining are from the IE department a) (10 points) What is the probability that 6 jobs from the IE department are sent to the printer in 25 minutes? b) (10 points) In the below table, you find 8 sets of random numbers uniformly distributed between (0, 1) generated by the RAND() function in Excel. Each set includes 3 numbers. Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6 Sample 7 Sample 8 0.9535 0.7081 0.6103 0.4896 0.0942 0.8250 0.9744 0.2400 0.7342 0.2385 0.7551 0.2553 0.6611 0.8930 0.2081 0.6645 0.8217 0.5055 0.2538 0.2207 0.6686 0.4537 0.6865 0.6404 Use these random numbers to calculate the empirical probability that 3 jobs from the EE department are sent to the printer in 10 minutes. For this purpose, you will follow the short algorithm outlined below: i) Generate exponentially distributed random numbers (with the appropriate rate) Hint: Recall that if we count the number of events in a certain time period under Poisson assumptions, then the times between the events are exponential. Add the three exponential numbers you generated in each sample (repeating this exercise i) Generate exponentially distributed random numbers (with the uppropriate rute) Hint: Recall that if we count the number of events in a certain time period under Poisson assumptions, then the times between the events are exponential. ii) Add the three exponential numbers you generated in each sample (repeating this exercise for each sample separately). Doing so, you will be calculating an occurrence time for the event that the third job from the EE department is sent to the printer. iii) In how many samples does the occurrence time exceed ten minutes? Divide that number by 8. This is the empirical probability c) (5 points) Calculate the probability that 3 jobs from the EE department are sent to the printer in 10 minutes (like in part a). Compare the empirical probability you calculated in part (b) with this theoretical result. Are these numbers close? What happens if you continue the exercise in part (b) with more data sets? Does the empirical probability get closer to your theoretical calculation of probability or further deviate from it? 2 Question 4) (25 points) On the third floor of the College of Engineering Building at the University of Western Arabia there is a printer used by the faculty of industrial engineering (IE) and electrical engineering (EE) departments. Print jobs are sent to the printer according to a Poisson distribution at a rate of 3 jobs per 5 minutes. An analysis of jobs that have been sent to the printer in the last year indicates that on average 60% of the jobs are from the EE department whereas the remaining are from the IE department a) (10 points) What is the probability that 6 jobs from the IE department are sent to the printer in 25 minutes? b) (10 points) In the below table, you find 8 sets of random numbers uniformly distributed between (0, 1) generated by the RAND() function in Excel. Each set includes 3 numbers. Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6 Sample 7 Sample 8 0.9535 0.7081 0.6103 0.4896 0.0942 0.8250 0.9744 0.2400 0.7342 0.2385 0.7551 0.2553 0.6611 0.8930 0.2081 0.6645 0.8217 0.5055 0.2538 0.2207 0.6686 0.4537 0.6865 0.6404 Use these random numbers to calculate the empirical probability that 3 jobs from the EE department are sent to the printer in 10 minutes. For this purpose, you will follow the short algorithm outlined below: i) Generate exponentially distributed random numbers (with the appropriate rate) Hint: Recall that if we count the number of events in a certain time period under Poisson assumptions, then the times between the events are exponential. Add the three exponential numbers you generated in each sample (repeating this exercise i) Generate exponentially distributed random numbers (with the uppropriate rute) Hint: Recall that if we count the number of events in a certain time period under Poisson assumptions, then the times between the events are exponential. ii) Add the three exponential numbers you generated in each sample (repeating this exercise for each sample separately). Doing so, you will be calculating an occurrence time for the event that the third job from the EE department is sent to the printer. iii) In how many samples does the occurrence time exceed ten minutes? Divide that number by 8. This is the empirical probability c) (5 points) Calculate the probability that 3 jobs from the EE department are sent to the printer in 10 minutes (like in part a). Compare the empirical probability you calculated in part (b) with this theoretical result. Are these numbers close? What happens if you continue the exercise in part (b) with more data sets? Does the empirical probability get closer to your theoretical calculation of probability or further deviate from it? 2
