MAST20006 Probability for Statistics /MAST90057 Elements of Probability
Assignment 5, Semester 1 2021
Due time: 4pm, Friday May 28.
Name:
Student ID:
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you have satisfied all of the above requirements.
MAST20006/90057 Semester 1, 2021 Assignment 5 2
• The submission deadline is 4pm Melbourne time on Friday 28 May. You have
longer than of the normal 10 days 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 (Rob
Maillardet).
• There are 6 assignment questions and 6 Maple project questions, of which some
randomly chosen questions will be marked. The assignment part is worth 4% of the
total mark and the Maple project is worth 10% of the total mark. Note that you are
expected to submit answers to all questions, otherwise a mark penalty will apply.
• Please attach your Maple worksheet at the end of your assignment. Your first line
of code should be your student ID number. If you need to refer to your Maple
attachment in your answers, please indicate clearly in your Maple worksheet which
question the answer/graph corresponds to.
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maplesoft.com/support/help/maple/view.aspx?path=worksheet%2Fmanaging%2FexportPDF
For instruction on how to merge your assignment PDF and the PDF of your Maple
worksheet into a single PDF document, use for example this online tool: https:
//smallpdf.com/merge-pdf
• Working and reasoning must be given to obtain full credit. Give clear and concise
explanations. Clarity, neatness, and style count.
∗ ∗ ∗
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MAST20006/90057 Semester 1, 2021 Assignment 5 1
1. Consider two random variables X and Y with the joint probability density
f(x, y) =
{
12xy(1− y), 0 < x < 1, 0 < y < 1
0 elsewhere.
Let Z = XY 2 and W = Y be a joint transformation of (X, Y ).
(a) Sketch the graph of the support of (Z,W ), and describe it mathematically.
(b) Find the inverse transformation.
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(c) Find the Jacobian of the inverse transformation.
(d) Find the joint pdf of (Z,W ).
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(e) Find the pdf of Z = XY 2 from the joint pdf of (Z,W ).
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2. Let X1, X2, · · · , Xn be a random sample of size n from a geometric distribution with
pmf
f(x) = 0.75 · 0.25x−1, x = 1, 2, 3, . . . .
(a) Find the mgf MY3(t) of Y3 = X1 +X2 +X3 using the geometric mgf. Then state
the distribution of Y3.
(b) Find the mgf MYn(t) of Yn = X1 +X2 + · · ·+Xn. Then state the distribution of
Yn.
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(c) Find the mgf MY¯n(t) of the sample mean Y¯n =
Yn
n
.
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(d) Find the limit limn→∞MY¯n(t) using the result of (c). What distribution does the
limiting mgf correspond to?
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(e) Let Zn =
3
2
√
n Y¯n− 2
√
n. Find MZn(t), the mgf of Zn. Then find the limiting mgf
limn→∞MZn(t). What is the limiting distribution of Zn?
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3. Let X1, X2 and X3 be a random sample of size n = 3 from the exponential distribution
with pdf f(x) = 1
2
e−x/2, 0 < x <∞. Find
(a) P (1 < X1 < 2, 2 < X2 < 3, 3 < X3 < 4).
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MAST20006/90057 Semester 1, 2021 Assignment 5 9
(b) E[X1X
2
2 (X3 − 2)2].
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4. Let X1, X2, X3 denote a random sample of size n = 3 from a distribution with the
Poisson pmf
f(x) =
2x
x!
e−2, x = 0, 1, 2, 3, . . . .
(a) Calculate P (X1 +X2 +X3 = 1) using only the known distributions of X1, X2, X3.
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(b) Find the moment-generating function of Z = X1 +X2 +X3 using the Poisson mgf
of X1. Then state the distribution of Z.
(c) Find the probability P (X1 +X2 +X3 = 4) using the result of (b).
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(d) If Y = max{X1, X2, X3}, find the probability P (Y ≤ 2).
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5. Let X be a gamma(θ = 3, α = 4) random variable. Use Chebyshev’s inequality to
determine
(a) A lower bound for P (5 < X < 19).
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(b) An upper bound for P (|X − 12| ≥ 18).
6. Suppose that the distribution of the weight of a prepackaged “1-kilo bag” of potatoes
is N(1.2, 0.052) and the distribution of the weight of a prepackaged “3-kilo bag” of
potatoes is N(3.3, 0.12). Now independently select at random three 1-kilo bags of
potatoes with weights being X1, X2 and X3 respectively. Let Y = X1 +X2 +X3. Also
randomly select one 3-kilo bag of potatoes with weight being W .
(a) Find the mgf of Y .
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(b) Find the distribution of Y , the total weight of the three 1-kilo bags of potatoes
selected.
(c) Find the probability P (Y > W ), i.e., the probability that the sum of weights of
three 1-kilo bags randomly selected is larger than the weight of one 3-kilo bag
randomly selected.
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MAPLE QUESTIONS
1. Please always simplify your final results in doing the following questions.
2. Use unassign(’obj’) if you want to remove an assigned obj.
3. Use restart to clean up your worksheet if it seems “messed up” and you get
lost.
4. After restart, don’t forget typing with(Statistics):
∗ ∗ ∗
1. There are one white marble and N−1 black marbles in a bag. Consider the experiment
of repeatedly picking out a marble at random from the bag and putting back a black
marble each time. Continue the experiment until the white marble is picked out.
Let X be a random variable equal k if the white marble is picked out at the k-step. It
can be shown that the pmf of X is
P (X = x) =
1
N
×
(
1− 1
N
)x−1
, x = 1, 2, 3, . . . .
Complete the following tasks and keep 3 significant digits after the decimal point in
your answers.
(a) Find the probability P (X ≥ 10). Simplify and factorize your result if possible.
(b) Find the mean E(X).
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MAST20006/90057 Semester 1, 2021 Assignment 5 17
(c) Find the variance Var(X).
(d) Find the mathematical expectation E[X exp(−X)].
2. Let a continuous random variable X have the following pdf
f(x) =
2
9
(x+ 1)(2− x), −1 < x < 2.
(a) Find the cdf F (x) of X.
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(b) Find the probability P (−3.5 < X < 1.8).
(c) Find the mean E(X).
(d) Find the mgf M(t) = E[exp(tX)].
(e) Find the third moment E(X3).
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(f) Let Y = X2. Find the range of Y and the pdf g(y) of Y .
3. Consider continuous random variables X and Y which have the following joint pdf
f(x, y) =
6
7
(x+ y)2, 0 < x < 1, 0 < y < 1.
(a) Find the marginal pdf f1(x) of X.
(b) Find the mean E(X).
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(c) Find the variance Var(X).
(d) Find the covariance Cov(X, Y ). (Note that E(X) = E(Y ).)
(e) Find the correlation coefficient ρ between X and Y . (Note that Var(Y ) =
Var(X).)
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(f) Find the probability P (2X + Y > 1).
4. Let X and Y be two independent exponential random variables with mean one, and
let Z = X/Y .
(a) Find the range of Z and the pdf f(z) of Z.
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(b) Use Maple to generate the graph of the pdf of Z for 0 ≤ z ≤ 10. Sketch the graph
by hand in the box below, and refer to your Maple attachment for the original
graph.
(c) What is the mean of Z?
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(d) What is the median of Z? What can you conclude about P (Z > 1)?
(e) What is P (Z > 3)?
(f) Let W = logZ. Find the mgf MW (t) of W . For which values of t is it well
defined?
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(g) From you result in (f), draw a conclusion on the existence of E[Zk] for a non-
negative integer k, and find E[
√
Z].
5. Let X be a continuous random variable with density function
f(x) =

1
x
, 1 < x < a
0 elsewhere
for some constant a.
For this question, we recall that the constant e := exp(1) must be defined as e:=exp(1)
in Maple. You may also need the function map(f,v) which evaluates a function f at
a vector v and returns a vector of values.
(a) Find the value of a.
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(b) Find the kth moment of X for k ≥ 1.
(c) Use Maple to plot the graph of the density f(x). Next, generate a sample of 5000
observations from the distribution of X and plot the corresponding histogram;
refer to your Maple attachment for the plots of the density and the histogram
(which you should try to show next to each other for comparison). In the box
below, write your comment on the comparison of the density with the histogram.
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(d) Find the cdf ofX and explain how you can use it to simulate a realisation/observation
from X.
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(e) Use (d) to generate another sample of 5000 observations from X, and give the
plot of the corresponding histogram in your Maple attachment. Compare this
histogram with the one you obtained in (d).
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(f) Let Z = ln(X). Find the mgf MZ(t) of Z. For which values of t is it well defined?
Compare its expression with your answer to (b) and comment.
(g) Find E[Z].
6. Let X be a chi-square random variable with mean one.
Let X1, X2, . . . , Xn be n independent random variables with the same distribution as
X. Let Yn =
∑n
i=1 Xi/n be the sample mean.
Plot the graphs of the pdf of X and the pdf of Yn for n = 3, 10, 20, 50 (use your
knowledge of the chi-square distribution to easily define Yn in Maple). Include these
graphs in your Maple attachment.
Answer the following questions in the answer box below: What do you observe when
comparing the graphs? Can you compare Yn to a known distribution when n is large?
Elaborate on your answers.
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