Mathematical Foundations of CmpE  Programming Assignment #3

ECE 3020 – Mathematical Foundations of CmpE 
Programming Assignment #3

You are required to complete this assignment INDIVIDUALLY.
Complete this programming assignment using a high-level procedural programming language (e.g., Java, C,
C++) of your choice, MATLAB, or an Excel spreadsheet (similar to the examples shown in lecture).
Carefully read the instructions below regarding printouts and other documentation; note that you do NOT
submit an executable copy of your program unless you implement your simulation using Excel.
The deliverables for this assignment must be submitted electronically on the Canvas “Assignments” page no
later than 11:00 pm on the due date listed above. A single pdf file is the preferred format. Clearly PRINT
your name at the top of each sheet. You are responsible for ensuring that your submitted file(s) are readable
and successfully uploaded to Canvas.
Probabilistic algorithms and/or stochastic simulation are often used to estimate values or predict the
likelihood of specific events, particularly if direct calculation is algorithmically complex or involves
interactions that are difficult to formally characterize, but easy to simulate computationally.
Consider the following “game” that uses a single fair die; i.e., the values 1 through 6 are equally likely for
each roll of the die: Starting with a total of zero (0), roll the die and add its value to the total, repeating
until the total of all the rolls first exceeds twelve (12). For example, the sequence of five rolls
6 – 2 – 1 – 2 – 5 results in a final total of 16.
(A) Before writing the program, answer the following questions, which may help you develop the
simulation. These questions can be answered by observation or simple reasoning, with no
significant calculations or computation needed.
i. What are the possible final totals? Why?
ii. Which final total is most likely? Why?
iii. What are the minimum and maximum number of rolls of the die required to achieve a valid
final total? Why?
(B) Develop a computational solution (program or spreadsheet) that simulates this game using a fair die
and estimates the following: for each possible final value, (i) what is the probability of achieving
this value and (ii) what is the average number of rolls required to achieve this value.
(C) Your simulation should generate 25,000 trials and produce the following outputs: (i) Results of the
first ten (10) trials listing the value of each die rolled, the final total, and the number of rolls. (This
will allow verification of your simulation.) (ii) A table summarizing the results (as defined in part B
above) after 25, 50, 100, 250, 500, 1000, 2500, 5000, 10000, 15000, 20000, and 25000 trials.
(D) Repeat the simulations from steps B and C, but with a non-uniform die that generates each of the
possible outcomes with the following specified probabilities:
Result: 1 2 3 4 5 6
Probability: 0.05 0.15 0.30 0.30 0.15 0.05
(E) Briefly summarize your observations from these simulations, including issues such as the number of
trials required for different values to converge, the stability of the results for various calculations,
and any other observations regarding the simulation process or results.
ECE 3020 – Mathematical Foundations of CmpE Spring 2019
Submit the following materials, preferably combined into a single .pdf file:
(a) Answers and brief explanations for the questions.
(b) Your program source code listing (.pdf or .txt) or Excel file (.xls).
(c) Program printout (or one-page portion of Excel spreadsheet) with the specified simulation results.
(d) Brief (1 or 2 paragraphs) written summary discussing the indicated issues.
Hints for generating random values to meet specific application needs:
Most programming languages and computing systems include the ability to generate random values,
typically in the form of a uniformly distributed real number between 0 and 1. By appropriately
scaling and shifting this value, a uniform distribution between any two real numbers can be easily
obtained. Non-uniform distributions, such as a Gaussian (normal) function, may be derived by
treating the system-generated value as the cumulative probability density and working backward to
calculate the desired function parameter.
Many applications require discrete random values distributed according to specified probabilities.
For equally likely values, the uniform random value can be scaled and then converted to an integer.
To avoid issues related to whether or not the 0 and 1 boundary values may occur, the scaled value can
be shifted by 0.5 and rounded to generate a series of equally likely integers. For example, the roll of a
single die can be simulated by the following calculation:
D = ROUND (6 * RAN() + 0.5)
which takes a random value in the range (0, 1), maps it to a value in the range (0.5, 6.5), and then
rounds it to an integer in the range [1, 6].
If an application requires discrete random values with unequal probabilities, values can be mapped
from a uniform space to a non-uniform space using CASE or IF-THEN-ELSE instructions or the
Excel HLOOKUP function. For example, uniform random values over (0, 1) can be mapped to the
Skee Ball distribution from Homework #5 as illustrated below:
Uniform (0, 1) random value
Skee Ball roll score
Spreadsheet programs like Excel often include built-in functions to generate random numbers
according to various sets of parameters, such as those discussed in the above paragraphs.
0.00
0.97
0.50 0.75 0.90 1.00
10
50
20 30 40