” MAST20004编程 写作、 辅导Python，CSSStudentNumberSemester 1 Assessment, 2020School of Mathematics and StatisticsMAST20004 ProbabilityThis exam consists of 27 pages (including this page)Authorised materials: printed one-sided copy of the Exam or the Masked Exam made availableearlier (or an offline electronic PDF reader), two double-sided A4 handwritten or typed sheetsof notes, and blank A4 paper.Calculators of any sort are NOT allowed.Instructions to StudentsYou should attempt all questions.There is a table of Normal distribution probabilities and Matlab output at the end of thisquestion paper.There are 9 questions with marks as shown. The total number of marks available is 120.Supplied by download for enrolled students only cUniversity of Melbourne 2020MAST20004 Probability Semester 1, 2020Question 1 (10 marks)Consider a random experiment with sample space.(a) Write down the axioms which must be satisfied by a probability mapping P defined onthe set of events of the experiment.(b) Using the axioms, prove that for any events A and B where B ? A, P(B) P(A).Page 3 of 27 pagesMAST20004 Probability Semester 1, 2020Question 2 (9 marks)A bag has three coins in it, one is fair, and the other two are weighted so the probability of ahead coming up are 512 and13 , respectively. You choose a coin at random from the bag and tossit.(a) What is the probability of a head showing on the coin?(b) Given a head is showing, What is the probability you tossed the fair coin?MAST20004 Probability Semester 1, 2020(c) Given a head is showing, if you toss the same coin again, what is the probability thatyou get a head?Do not attempt to calculate the final answer. Leave your answer in terms ofproducts and quotients of fractions.Page 5 of 27 pagesMAST20004 Probability Semester 1, 2020Question 3 (12 marks)The Weibull cumulative distribution function FX is given byFX(x) ={1 e(x/) , x 00, otherwise,where 0 and 0 Are parameters.(a) How could a realisation of a Weibull random variable be generated from an R(0, 1)random number generator?Page 6 of 27 pagesMAST20004 Probability Semester 1, 2020(b) Let Y = X . Derive the cumulative distribution function of Y , identify the distributionof Y , and write down E(Y ).(c) Let Z = min(Y,M) Where M is a positive finite number. Using the tail probabilitiesformula for the mean, derive an expression for E(Z).Page 7 of 27 pagesMAST20004 Probability Semester 1, 2020Question 4 (19 marks)Let X and Y have joint probability density function given byf(X,Y )(x, y) ={cxy, 0 y x 20, otherwisewhere c is a constant.(a) Plot the region where f(X,Y )(x, y) is nonzero.Page 8 of 27 pagesMAST20004 Probability Semester 1, 2020(b) Find the constant c.(c) Find fY (y), the marginal Probability density function of Y .Page 9 of 27 pagesMAST20004 Probability Semester 1, 2020(d) Find fX|Y (x|y), the conditional probability density function of X given Y = y.(e) Evaluate P请加QQ：99515681 或邮箱：99515681@qq.com WX：codehelp
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