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MAST20004 Probability

Semester 2, 2024

Assignment 3: Questions

Due 4 pm, Wednesday 18 September

Important instructions:

(1) This assignment contains 4 questions, two of which will be randomly selected to be marked. Each marked question is worth 10 points and each unmarked question with substantial working is worth 1 point.

(2) 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.

• If you do not have a printer but you can figure out how to annotate a PDF using 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.

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’.

(3) A poor presentation penalty of 10% of the total available marks 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.

(4) Late submission within 20 hours after the deadline will be penalised by 5% of the to-tal 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. You are strongly encouraged to submit the assignment a few days before the deadline just in case of unexpected technical issues. If you are facing a rather excep-tional/extreme situation that prevents you from submitting on time, please complete the assignment extension request form that is available on Canvas.

(5) Working and reasoning must be given to obtain full credit. Clarity, neatness, and style count.

Q1. Let X be a random variable with PDF

(a) Put Y1 = X3 . Derive the CDF and PDF of Y1.

(b) Put Y2 = X2 . Derive the CDF and PDF of Y2.

(c) Put Y3 = min{X, 1}. Derive the CDF and PDF of Y3.

(d) Let U ∼ Unif (0, 1). Explain theoretically how to generate a realisation of the RV X from U.

(e) Put U = runif (1000). Using your answer from part (d), write an R code to simulate 1000 realisations of X from U.

Q2. Let (X, Y ) be a discrete random vector with joint PMF

(a) Sketch the region for which pX,Y (i, j) > 0 (with horizontal i-axis and vertical j-axis).

(b) Compute the marginal PMF of X.

(c) Compute the marginal PMF of Y .

(d) Determine the conditional PMF of X given Y = j.

(e) Are X and Y independent? Explain.

(f) Compute the PMF of the random variable V := Y − X.

Q3. Let (X, Y ) be a continuous random vector with joint PDF

where C > 0 is a constant.

(a) Sketch the region for which fX,Y (x, y) > 0 (with horizontal x-axis and vertical y-axis).

(b) Find the value of C.

(c) Find the marginal PDFs of X and Y .

(d) Are X and Y independent? Explain.

Q4. Let X and Y be random variables on a common probability space (Ω, P). Determine whether each of the following statements is true or false. If it is true, give a proof or an explanation by quoting basic properties or facts from the lecture slides/tutorials; if it is false, give a counterexample or an explanation.

(a) If X  −X, then E(X) = 0.

(b) If X  exp(λ) and Y = b Xc , then E(Y ) = e −λ/(1 − e −λ ).

(c) If X  exp(λ) and Y = max{X, 10}, then the distribution FY satisfies the mem-oryless property.

(d) If X is a random variable with P(X ≥ 0) = 1 and E