UNIVERSITY OF TORONTO Faculty of Arts and Science AUGUST 2016 EXAMINATIONS STA347H1S Duration - 2 hours Examination Aids: A non-programmable calculator Last name: First name: Student number: Middle name: Instruction 1. If a question asks you to do some calculations or derivations, you must show your work to receive full credit. Simplify expressions as much as you can. 2. If you do not have enough space, use the other side and refer it correctly. 3. Some problems could be wrong. If it happens, correct questions by yourself after arguing why the problem is wrong. 4. Please do not pull the pages apart unless necessary. If you do so, print your name and sign all the pages. 5. You can use either pen or pencil for the test. But please be aware that you are not allowed to dispute any credit after the test is returned if you use pencil. 6. You are allowed to use any theorem covered in class as well as any statement in other questions in this exam. For example, in question x (y), you can use the result of question x (z) without a proof. 7. There are 7 questions on 3 sheets, and the total is 100 marks. Please hand in your exam paper and books when the time is up. 8. Some useful information are on page 3. 1 Problem 1. Suppose that X is a random variable of which moment generating function is mgfX (t) = (1 + 3et + et + e2t )/6 for < t < . (a) [10 marks] Find the mean and variance of X. (b) [5 marks] Compute P (X > 0). Problem 2. [10 marks] Two independent random variables X and Y satisfy X gamma(8, 4) and X + Y gamma(10, 4). Find the characteristic function of Y . Problem 3. [15 marks] Two random variables X and Y satisfy E(X | Y ) = 20 + 0.8Y and E(Y | X) = 16 + 0.1X. Compute E(X), E(Y ) and Corr(X, Y ) when X and Y have finite second moments. Problem 4. Two random variables X and Y have joint probability density function given by pdfX,Y (x, y) = cxy1(0 < x < y < 1). (a) [5 marks] Find the value of constant c. (b) [5 marks] Compute marginal density of X and its expectation. (c) [5 marks] Prove or disprove X and Y have the same distribution. (d) [5 marks] Prove or disprove X and Y are independent. Problem 5. New hypothesis is propose in the process of a storm development. A storm is generated as unstable status (U). As time goes its status changes to disappear (D) or normal storm (N) or strong storm (S) or remains unstable. The status of the development process follows a Markov chain having transition matrix D p=U N S D 1.0 0.2 0.0 0.0 U 0.0 0.4 0.0 0.0 N 0.0 0.3 0.8 0.6 S 0.0 0.1 0.2 0.4 (a) [5 marks] Determine whether or not each state is transient or recurrent. (b) [5 marks] Compute the limit probability of each status when it started unstable status, that is, lim p(n) (U, s) each state s = D, U, N, S. n (c) [5 marks] Compute the expected time for U absorbed into recurrent states. Problem 6. Suppose Yi ind. Poisson(i ) where i = xi where is an unknown positive number and xi 's are known positive numbers satisfying lim (xk1 + + xkn )/n mk > 0 for n Qn b k = 1, 2, 3, 4. Let = argmax i=1 pmfYi (yi ). P P (a) [5 marks] Show that b = ni=1 Yi / ni=1 xi . (b) [5 marks] Show that b converges to in probability. (c) [5 marks] Show that n(b ) converges to N (0, 2 ) in distribution. State 2 using and mk 's. 2 Problem 7. [10 marks] Pair-wise independent random variables X1 , X2 , . . . satisfy E(Xn ) = for all n and lim Var(Xn ) = 0. Show that X n = (X1 + + Xn )/n in distribution. n Useful Information 1. Probability mass/density functions Distribution pmf/pdf Bernoulli(p) px\u0001(1 p)1x n x binomial(n, p) p (1 p)nx x x Poisson() e /x! x1 geometric(p) p(1 \u0001 p) x1 k negbinomial(k, p) k1 p (1 p)xk uniform(a, b) 1/(b a) exponential() ex gamma(, ) (())1 ex x1 (+) beta(, ) x1 (1 x)1 ()/() N (, 2 ) 2 (2 2 )1/2 exp( (x) ) 2 2 2. The gamma function is defined by () = ( + 1) = (). 3. The sum of geometric sequence is P domain x = 0, 1 x = 0, . . . , n x = 0, 1, 2, . . . x = 1, 2, 3, . . . x = k, k + 1, . . . a0 x>0 0 0 and satisfies n=0 n mgf 1 p + pet (1 p + pet )n t e(e 1) pet /(1 (1 p)et ) [pet /(1 (1 p)et )]k