代做MAST20004 Probability Semester 2, 2024 Assignment 3: Questions代做留学生R程序
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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.
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(3) A poor presentation penalty of 10% of the total available marks will apply unless your submitted assignment meets all of the following requirements:
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(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