MIDTERM TEST -- CSC 354S University of Toronto Spring 1995 50 marks -- 50 minutes Test aids allowed: one 8.5" X 11" (double-sided) fact sheet, non-programmable scientific/statistical calculator, ruler 9Notes to students: State all assumptions where appropriate. In general, show your intermediate work. 1. [[[[22220000 mmmmaaaarrrrkkkkssss ttttoooottttaaaallll]]]] 9Consider the function _f(_x) = 20_x(1-_x)7399 , 0 < _x < 1 . 9a) [[[[6666 mmmmaaaarrrrkkkkssss]]]] Show that _f is indeed a density function. 9b) [[[[4444 mmmmaaaarrrrkkkkssss]]]] Give the formula for the distribution function _F that corresponds to the density function _f. 9c) [[[[11110000 mmmmaaaarrrrkkkkssss]]]] Using the acceptance-rejection method, show how to generate efficiently random variates having the den- sity function _f. Be explicit; show details of your work. 2. [[[[8888 mmmmaaaarrrrkkkkssss]]]] 9Explain succinctly in a few sentences what the basic idea (purpose, principle) of maximum-likelihood estimators is. 9[Content, meaning correctness, is of course important in this answer, but style, meaning pithiness, is as well. State it correctly and state it simply.] 3. [[[[11110000 mmmmaaaarrrrkkkkssss]]]] 9In Problem Set 2, question 2, you were to implement an _M/_M/1 queue to calculate delay times in queue, based on a formula. Specifically, there were IID interarrival times _A918, _A928, . . . and IID service times _S918, _S928, . . . . Customers were served in a FIFO manner. This gave rise to a sequence of delays in queue _D918, _D928, . . . . Justify the formula that related _D9_i+18 to quantities in the sequences {_A}, {_S}, and {_D}, and to certain constants. 4. [[[[11112222 mmmmaaaarrrrkkkkssss ttttoooottttaaaallll]]]] 9Suppose we have a system that admits to a terminating simu- lation. We make 5 independent replications and construct a point estimate and 95 percent confidence interval for the mean throughput rate, based on the obtained throughput values _X918 = 7221, _X928 = 7155, _X938 = 7250, _X948 = 7260, and _X958 = 7260. Note that page 2 of this test contains a useful sta- tistical table. 9a) [[[[2222 mmmmaaaarrrrkkkkssss]]]] Calculate the point estimate and the confidence interval. 9b) [[[[5555 mmmmaaaarrrrkkkkssss]]]] Approximately how many replications in total 9 February 17, 1997 .../continued - 2 - would be required to obtain an absolute error of 50? 9c) [[[[5555 mmmmaaaarrrrkkkkssss]]]] Approximately how many replications in total would be required to obtain a relative error of 5 percent? EEEEnnnndddd ooooffff tttteeeesssstttt (but remember about page 2) TTTToooottttaaaallll MMMMaaaarrrrkkkkssss ==== 55550000 9 February 17, 1997 .../continued - 3 - February 17, 1997