ESE 503 - Simulation Modeling & Analysis (Optional Take-Home Poblem)
Spring Semester, 2022
Instructions:
1.) You may do this problem (it is optional) and upload it to Canvas by 11:59 PM (EDT) on Wednesday, March 16th, 2022.
2.) Since the total points in the in-class portion of the exam added to 100, this is an optional extra credit assignment and so no late assignment will be accepted, so do not wait until the last minute to upload your assignment (no exceptions to this rule).
3.) This will be used to replaced your lowest problem grade in your in-class midterm with this grade only if it makes your final midterm grade higher, so if the five problems you did in the midterm had grades of A, B, C, D and E (all out of 20 points each) and if your take-home problem grade is F (out of 20 points) then the following formula will be used for your final midterm grade.
G = A + B + C + D + E + F − min(A,B,C,D,E,F)
4.) You must upload your theoretical calculation as a pdf file and your simulation should be an Excel file or any other program of your choosing.
5.) Please include this page with your signature testifying to the fact that you did this problem on your own without the use of the internet or help from anyone else. Without this, you problem will not be graded.
Print Your Name: __________________________________________________
Sign Your Name: ___________________________________________________
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Take-Home Optional Problem (20 points) - Computing PDF and CDF Functions
Suppose that X ~ U[0, 1) and Y ~ U[0, 1) and that Z = X2 + 2Y.
a.) (10 points) Determine the pdf (f) and cdf (F) for Z and for full credit, you must write your final answer without the need for min and max functions.
b.) (4 points) Make plots of both f(z) and F(z) as functions of z.
c.) (6 points) Test your result by running a 10, 000-sample simulation showing the simulated values of
P(0 < Z < 1/3) P(1/3 < Z < 2/3) P(2/3 < Z < 1)
P(1 < Z < 4/3) P(4/3 < Z < 5/3) P(5/3 < Z < 2)
P(2 < Z < 7/3) P(7/3 < Z < 8/3) P(8/3 < Z < 3)
and the theoretical values that you must compute and also show the two results side by side on a histogram plot.
Solution to Take-Home Problem #1
a.) First we note that 0 < Z < 3 and then we use
c.) The cdf F(z) can now be used to compute
P(z1 < Z < z2) = F(z2) − F(z1)
and these are shown in the Excel file midterm_ths.xlsx provided on Canvas. The
agreement between theory and simulation is very good.