MAST20006 Probability for Statistics /MAST90057 Elements of Probability

Assignment 5, Semester 1 2021

Due time: 4pm, Friday May 28.

Name:

Student ID:

• To complete this assignment, you need to write your solutions into the blank answer

spaces following each question in this assignment PDF.

If you have a printer (or can access one), then you must print out the assignment

template and handwrite your solutions into the answer spaces.

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an iPad/Android tablet/Graphics tablet or using Adobe Acrobat, then annotate your

answers directly onto the assignment PDF and save a copy for submission.

Failing both of these methods, you may handwrite your answers as normal on blank

paper and then scan for submission (but note that you will thereby miss valuable

practice for the exam process). In that case, however, your document should have

the same length as the assignment template otherwise Gradescope will reject your

submission. So you will need to add as many blank pages as necessary to reach that

criterion.

Scan your assignment to a PDF file using your mobile phone (we recommend Cam-

Scanner App), then upload by going to the Assignments menu on Canvas and submit

the PDF to the GradeScope tool by first selecting your PDF file and then clicking on

‘Upload PDF’.

• A poor presentation penalty of 10% will apply unless your submitted assignment

meets all of the following requirements:

– it is a single pdf with all pages in correct template order and the correct way up,

and with any blank pages with additional working added only at the end of the

template pages;

– has all pages clearly readable;

– has all pages cropped to the A4 borders of the original page and is imaged from

directly above to avoid excessive ’keystoning’

These requirements are easy to meet if you use a scanning app on your phone and take

some care with your submission - please review it before submitting to double check

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.

For instructions on how to save a Maple worksheet as a PDF file, visit https://www.

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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MAST20006/90057 Semester 1, 2021 Assignment 5 2

(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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(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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