MATH20722 Two hours THE UNIVERSITY OF MANCHESTER FOUNDATIONS OF MODERN PROBABILITY Answer THREE of the FOUR questions. If more than 3 questions are attempted, credit will be given for the best 3 answers. Each question is worth 20 marks. Electronic calculators may be used, provided that they cannot store text. 1 of 5 P.T.O. 04 June 2018 14.00 - 16.00 © The University of Manchester, 2018. MATH20722 Answer 3 questions 1. a) i) Dene a eld F of a set . ii) Let F be a eld of and suppose B 2 F . Show that G = fA \B : A 2 Fg is a eld of B. iii) Let F and L be elds of : Let F \ L be the collection of subsets of lying in both F and L. Show that F\L is a eld of . iv) Let F and L be elds of . Let F [ L be the collection of subsets of lying in at least one of F and L. Show, by means of a counter-example, that F [ L is not necessarily a eld of . [11 marks] b) i) State the Cauchy-Schwarz inequality. ii) By considering the existence of roots of the quadratic equation E[(X+xY )2] = 0 in x, or otherwise, prove the Cauchy-Schwarz inequality. iii) Let Y be a non negative random variable with 0 < E[Y 2] < 1. Consider the random variables Y and IfY >0g and use the Cauchy-Schwarz inequality to conclude that P (Y > 0) E[Y ] 2 E[Y 2] : [9 marks] 2 of 5 P.T.O. MATH20722 2. a) Let (Xn)n1 be a sequence of independent random variables such that P (Xn = n) = 1=n P (Xn = 1) = 1 2=n P (Xn = 2) = 1=n: For each of the following modes of convergence, determine, giving full explanations, whether Xn converges as n!1: i) convergence in mean; ii) convergence in probability; iii) convergence almost surely. [8 marks] b) Let (Xn)n1 be independent random variables, each having probability density function f(x) = 1 x2 x 1 0 otherwise: Determine P (En innitely often) for each of i) En = fXn > n 32g ii) En = fX4 > p ng iii) En = fX4 > 2 + 1ng iv) E2n = E2n+1 = fX2n > ng [12 marks] 3 of 5 P.T.O. MATH20722 3. a) Given random variables (Xn)n1; (Yn)n1; X and Y , dened on a common probability space and a sequence (cn)n1 of real numbers converging to c, show that: i) if Xn ! X and Yn ! Y , both in probability, then Xn + Yn ! X + Y in probability; ii) if Xn ! X in probability then cnXn ! cX in probability; iii) if Xn ! c in distribution, then Xn ! c in probability. [12 marks] b) Let (Xn)n1 be independent, identically distributed random variables, each having mean 0 and nite fourth moment. Dene Xn = Pn k=1Xk n : Show that, as n!1, Xn ! 0 almost surely: [8 marks] 4 of 5 P.T.O. MATH20722 4. a) i) Dene the term "characteristic function". ii) Show that the characteristic function of a random variable X is real valued if and only if X is symmetrically distributed about zero. iii) (Xn)n1 are independent and identically distributed random variables, each having characteristic function eajtj, where a is a real positive constant. Show thatPn k=1Xk n converges in distribution to some random variable Z. Explain why Z is not a constant and why the result does not violate the weak law of large numbers. [10 marks] b) i) For > 0, X is a Poisson random variable with distribution given by P (X = k) = ke k! k = 0; 1; ::: Show that the characteristic function of X is expf(eit 1)g: Carefully stating any results to which you appeal, show that, as !1, the random variable X p converges in distribution to a random variable X, which has the standard normal distribution. ii) Show that, as n!1, nX i=0 nien i! ! 1=2: [10 marks] END OF EXAMINATION PAPER 5 of 5
























































































































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