4580/5580/5581 ORIE Homework Assignment 5
Due by Oct. 7, 11:00 am, in the drop-box located on the second floor of Rhodes Hall. Please show your work and clearly mark your answers. Make sure that your spreadsheet printouts are properly formatted and labeled. There is no need to print out the entire sheet. The first page suffices, as long as the confidence interval calculations are shown. Hand in a printout of the formulas on your spreadsheets as well. In Excel, CTRL` key combination can be used to display the formulas (` key is located at the top left corner of the keyboard). Do not forget to indicate a recitation session on your homework. This is the session during which you can collect your graded work. 1. The p.d.f. of the random variable X is given by [ 3 -x3 + 4x2 + x - 4] if 1 x 3 32 f (x) = 0 otherwise. a) Sketch this p.d.f.. (You can use Excel.) b) Give an acceptance-rejection algorithm to generate samples of X. c) On the average, how many samples from the uniform distribution over [0, 1] would your acceptance-rejection method need in order to generate one sample of X? d) By using your acceptance-rejection method, generate 500 samples of X. Plot a histogram of the samples and compare it with the p.d.f. you sketched in Part a.
2. Assume that the arrivals a to coffee shop can be modeled by using a nonstationary Poisson process with rate function (t) = 3 2 t - 2t + 10 for 0 t 12, 10
where t is measured in hours and t = 0 corresponds to 8 : 00 am. a) Sketch the arrival rate function over the time interval [0, 12] (i.e., from 8:00 am to 8:00 pm). Compute the maximum value of (t) over the time interval [0, 12]. Let be this maximum. b) Compute the expected number of arrivals to the shop over the time interval [4, 8] (i.e., from 12:00 pm to 4:00 pm). c) Use simulation to build a 95% confidence interval for the expected number of arrivals over the time interval [4, 8]. In particular, (i) generate the arrival times of first 1500 customers in a stationary Poisson process with arrival rate ; (ii) use the method discussed in class to 1
generate the arrival times of the nonstationary Poisson process with arrival rate function (); (iii) count the total number of arrivals over the time interval [4, 8]; (iv) pressing F9 multiple times, collect 30 or so samples of the total number of arrivals during the time interval [4, 8] (you can also use @Risk for this purpose); (v) build a confidence interval based on the 30 or so samples. (Hint: 1500 is large enough so that the 1500-th customer will almost surely arrive after t = 8).
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