MAST20006 Probability for Statistics /MAST90057 Elements of Probability
Assignment 4, Semester 1 2022
Due time: 4pm, Friday 13 May.
Name:
Student ID:
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MAST20006/90057 Semester 1, 2022 Assignment 4 1
• The submission deadline is 4pm Melbourne time on Friday 13 May. You have
longer than of the normal one week to complete this assignment. Late submission
within 20 hours after the deadline will be penalised by 5% of the total available
marks for every hour or part thereof after the deadline. After that, the Gradescope
submission channel will be closed, and your submission will no longer be accepted. We
recommend you submit at least a day before the due date to avoid any technical delays.
If there are extenuating, eg medical, circumstances, contact the Tutorial Coordinator.
• There are 5 questions, of which 2 randomly chosen questions will be marked. Note
you are expected to submit answers to all questions, otherwise a mark penalty will
apply.
• Working and reasoning must be given to obtain full credit. Give clear and concise
explanations. Clarity, neatness, and style count.
∗ ∗ ∗
1
MAST20006/90057 Semester 1, 2022 Assignment 4 2
1. A random variable X has a probability density function given by
fX(x) =
2x+ 6
25
, −3 < x < 2.
Let Y = |X|.
(a) What is SY ?
(b) Is the transformation one-to-one?
2
MAST20006/90057 Semester 1, 2022 Assignment 4 3
(c) Derive the cumulative distribution function of Y .
3
MAST20006/90057 Semester 1, 2022 Assignment 4 4
(d) Derive the probability density function of Y .
4
MAST20006/90057 Semester 1, 2022 Assignment 4 5
2. Let (X, Y ) be a continuous bivariate random variable having the joint probability
density function
f(x, y) = cxy, 0 ≤ x ≤ y ≤ 2
for some real constant c.
(a) Sketch the graph of the support of (X, Y ).
5
MAST20006/90057 Semester 1, 2022 Assignment 4 6
(b) Find the value of c.
(c) Find fX(x), the marginal probability density function of X.
6
MAST20006/90057 Semester 1, 2022 Assignment 4 7
(d) Find fY (y), the marginal probability density function of Y .
7
MAST20006/90057 Semester 1, 2022 Assignment 4 8
(e) Compute µX , µY , σ
2
X , σ
2
Y ,Cov(X, Y ), and ρ.
8
MAST20006/90057 Semester 1, 2022 Assignment 4 9
(f) Find g(y|x = 1
2
), the conditional probability density function of Y given X = 1
2
.
(g) Find P
(
Y > 3
4
|X = 1
2
)
.
9
MAST20006/90057 Semester 1, 2022 Assignment 4 10
(h) Find P
(
X < 1|Y < 3
2
)
.
10
MAST20006/90057 Semester 1, 2022 Assignment 4 11
3. Let (X, Y ) be a bivariate random variable, and let a and b be real constants.
Show that
(a) Cov(aX, bY ) = abCov(X, Y ).
(b) Cov(X + a, Y + b) = Cov(X, Y ).
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MAST20006/90057 Semester 1, 2022 Assignment 4 12
(c) Cov(X, aX + b) = aVar(X).
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MAST20006/90057 Semester 1, 2022 Assignment 4 13
4. Let X and Y have the joint pmf defined by f(0, 0) = f(1, 2) = 0.2, f(0, 1) = f(1, 1) =
0.3 (and f(x, y) = 0 elsewhere).
(a) Represent the joint probability mass function by a table.
(b) Give the marginal probability mass functions of X and Y in the “margins” of the
table in (a).
13
MAST20006/90057 Semester 1, 2022 Assignment 4 14
(c) Compute µX , µY , σ
2
X , σ
2
Y ,Cov(X, Y ), and ρ.
14
MAST20006/90057 Semester 1, 2022 Assignment 4 15
(d) Tabulate the conditional probability mass function of Y given X = 1.
(e) Tabulate the conditional probability mass function of X given Y = 0.
15
MAST20006/90057 Semester 1, 2022 Assignment 4 16
5. Let X have a uniform distribution on the interval (0, 2). Given X = x, let Y have a
(conditional) uniform distribution on the interval (0, 2x).
(a) Write down the conditional probability density function of Y given that X = x.
Make sure to include the support of Y for each possible value of x.
(b) Find E(Y |x) and E(Y ).
16
MAST20006/90057 Semester 1, 2022 Assignment 4 17
(c) Find Var(Y |x) and Var(Y ).
17
MAST20006/90057 Semester 1, 2022 Assignment 4 18
(d) Determine f(x, y), the joint probability density function of (X, Y ).
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