Two hours. Statistical Tables to be provided THE UNIVERSITY OF MANCHESTER. 14 January :45 11:45

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1 Two hours Statistical Tables to be provided THE UNIVERSITY OF MANCHESTER PROBABILITY 2 14 January :45 11:45 Answer ALL four questions in Section A (40 marks in total) and TWO of the THREE questions in Section B (40 marks in total). If more than TWO questions from Section B are attempted then credit will be given for the best TWO answers. Electronic calculators are permitted, provided that they cannot store text. 1 of 5 P.T.O.

2 SECTION A Answer ALL of the FOUR questions A1. Let X be a discrete random variable such that X P o(λ) where λ > 0, that is the probability mass function of X is given by p X (k) = λk k! e λ, for k = 0, 1, 2,... (a) Prove that the probability generating function of X is given by P X (t) = e λ(t 1) for t R. (b) Hence find the mean and variance of X. A2. Suppose that (X, Y ) is a bivariate continuous random variable with joint probability density function { c(y f X,Y (x, y) = 2 x 2 ) for 0 < x < y < 1, 0 otherwise. (a) Prove that c = 6. (b) Find P(X + Y > 1). (c) Find P(X > 1/2 X + Y > 1). A3. Let X and Y be independent random variables with probability density functions f X (x) = 1, x (0, 1) and 0 otherwise; f Y (y) = 2y, y (0, 1) and 0 otherwise. (a) Derive the probability density function of Z 1 = min(x, Y ); (b) Derive the probability density function of Z 2 = 1/Y 2. A4. Chebyshev s Inequality states that, if X is a random variable with mean E[X] = µ and variance var(x) = σ 2 then P( X µ rσ) 1/r 2, for all r > 0. 2 of 5 P.T.O.

3 (a) Prove that if X has mean E[X] = 3 and variance var(x) = 4, P( 3 < X < 9) 8/9. (b) Let X 1,..., X n be a sequence of independent identically distributed random variables with unknown mean E[X 1 ] = µ and variance var(x) = σ 2 = 4. Find the mean and variance of the sample mean X n. (c) By applying Chebyshev s inequality to X n, determine a value of n so that P( X n µ < 0.5) > of 5 P.T.O.

4 SECTION B Answer TWO of the THREE questions B5. Suppose that (X, Y ) is a bivariate continuous random variable with joint probability density function { 14xy 4 for 0 < x < y < 1, f X,Y (x, y) = 0 otherwise. (a) Find the marginal probability density functions of X and of Y ; (b) Find cov(x, Y ); (c) Find the conditional probability density function f X Y (x y) stating for which values of y it is defined; (d) Find P(X > 1/3 Y = 2/3); (e) Find E[X Y = 2/3]; (f) confirm that E[E[X Y ]] = E[X] by evaluating each side from its definition. B6. Let X be a continuous random variable such that X N(µ, σ 2 ), that is the probability density function of X is given by f X (x) = 1 ) (x µ)2 exp (, x R. 2πσ 2σ 2 (a) Prove that the moment generating function of X is given by M X (t) = exp (µt + σ2 t 2 2 [Hint: You can use without proof the standard integral 2 dv = 2π] (b) Hence find the mean and variance of X. e v 2 (c) Suppose that X i (1 i n) are independent random variables with moment generating functions M Xi (t) and Y = X X n. Prove that the moment generating function of Y is determined by the moment generating functions of the X i as follows: M Y (t) = n i=1 M X i (t). (d) Let X 1, X 2,... X n be n independent continuous random variables with X i N(µ i, σ 2 i ) and let Y = X X n. Find the distribution of Y. (e) Let X N(0, 1) and Y N(0, 1) be independent of each other. Let U = (X + Y )/ 2, V = (X Y )/ 2. Find the joint probability density function f U,V (u, v). ). 4 of 5 P.T.O.

5 B7. (a) State the Central Limit Theorem. When a real number is rounded off to the nearest integer the difference is called the rounding error. When n real numbers are added and the resulting integers added the rounding errors accumulate to give a total rounding error. Assume that the rounding errors are independent and that the errors are uniformly distributed on the open interval ( 1/2, 1/2). (b) Find the probability density function for the rounding error when two real numbers are rounded off and added. [Recall the convolution formula: f Z (z) = f X(x)f Y (z x)dx when Z = X+Y is the sum of two independent continuous random variables.] (c) Explain how the Central Limit Theorem can be used to estimate the distribution of the total rounding error when n real numbers are rounded and added for large n. (d) What is the probability that the total rounding error is less than 10 when 4800 real numbers are rounded and added? (e) Let p n be the probability that the total rounding error is less than 12 when n real numbers are rounded and added. What is the maximum number of real numbers which may be rounded and added so that p n 0.9? END OF EXAMINATION PAPER 5 of 5