ECMT2160: Computational

ECMT2160: ComputationalAssignment
Due: November 3, 11:59am
This assessment task requires you to use MATLAB to run some Monte Carlo
simulations. You should prepare your submission as a MATLAB Live Script
file (i.e., a .mlx file). Submit your answers through the Canvas course website.
Your submission should include a mixture of written responses formatted as text,
blocks of MATLAB code, and MATLAB output, including graphs. You should
submit two versions of your answers: the original .mlx file, and a version ex-
ported to .html.
You may work on this assessment individually, or in pairs. If you work in
pairs, it is important that you clearly indicate the student ID number of your
partner in your submission. Your submission should not be identical to your
partner’s submission.
The assignment consists of two questions, each with multiple parts. Answer
all parts of both questions. The assignment is worth a total of 25 points towards
your final assessment. The first question is worth 15 points and the second ques-
tion is worth 10 points. Points will be deducted for poor presentation, including:
excessive typos, poor written expression, poor organization, etcetera.
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Question 1
Before attempting Question 1, you should work through Sections 3.8 and 3.9 in
the file IntroProb.mlx. Begin your submitted solution to Question 1 by running
the command
rng(STUDENTID)
in MATLAB, where STUDENTID is your 9-digit Student ID number. This fixes
the sequence of random numbers to be generated in your simulation.
Suppose we roll two fair six sided dice. Let1 denote the sum of the numbers
rolled, and let 2 denote the maximum of the numbers rolled.
(a) (i) Create an 11 × 6 matrix containing the values taken by the joint prob-
ability mass function of 1 and 2. The entry in row , column of this
matrix should contain the probability P(1 = , 2 = ).
(ii) Create a three-dimensional bar graph displaying the joint probability
mass function of 1 and 2.
(b) (i) Create a 1× 6 vector containing the values taken by the marginal prob-
ability mass function of 2. The entry in column of this vector should
contain the probability P(2 = ).
(ii) Create a two-dimensional bar graph displaying themarginal probability
mass function of 2.
(c) (i) Create an 11 × 6 matrix containing the values taken by the conditional
probabilitymass function of1 given2. The entry in row , column of
this matrix should contain the conditional probability P(1 = |2 = ).
(ii) Create six two-dimensional bar graphs, each displaying the conditional
probability mass function of 1 given 2 = , with taking the values
1 through 6 in your six graphs.
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(d) In each of 10,000 iterations of a “for loop”, do the following.
(i) Generate a discrete random variable whose probability mass function
is the marginal probability mass function of 2 calculated in part (b).
Hint: theMATLAB commandrandi(6,2,1) returns a 2×1 random
vector whose entries are independent random variables each equal to
the numbers 1 through 6 with equal probabilities.
(ii) Calculate the conditional expectation E(1 | 2 = ), where is the
random number generated in part (i).
Calculate the average of the conditional expectations computed over all 10,000
iterations of the “for loop”.
(e) Discuss how your findings in part (d) relate to the Law of Iterated Expecta-
tions.
3
Question 2
Let be a random variable with the standard normal distribution, and let() be
the probability density function of the standard normal distribution. Let (1, 2)
be the function
(1, 2) =
{
4(12)?4(?31 + ?32 ? 1)?7/3 if 0 < 1 < 1 and 0 < 2 < 1
0 otherwise.
Suppose that 1 and 2 are a pair of continuous random variables whose joint
probability density function is given by
(1, 2) = (P( ≤ 1), P( ≤ 2))(1)(2)
for all real 1 and 2.
(a) Create a graph of the joint probability density function of1 and2 for values
of 1 and 2 between ?3 and 3.
(b) Create a graph of the marginal probability density function of 1 for values
of 1 between ?3 and 3. Graph it alongside the standard normal probability
density function.
(c) Repeat part (b) for 2 instead of 1.
(d) Based on your answers above, do you think that the joint distribution of 1
and 2 is multivariate normal? Why or why not?