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LANCASTER UNIVERSITY
2012 EXAMINATIONS
PART II (Third or Fourth Year) MATHEMATICS & STATISTICS 2 Hours MATH 313/MATH413: Probability and Measure
You should answer ALL Section A questions and TWO Section B questions. In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40. SECTION A
A1. Let Ω = { 0 , 1 , 2 ,... } and let F be the set of all subsets of Ω. (i) Prove that F is a σ-algebra. [4] (ii) Let 0 < p < 1 and let P({j}) = p(1 − p)j^ for j ∈ Ω. Prove that P determines a unique probability measure on Ω. [6] (iii) Let X : Ω → R be given by X(j) = j for all j ∈ Ω. Verify that X is a random variable. Calculate and sketch the cumulative distribution function F of X. [6]
A2. (i) Let X be a random variable with probability density function
p(x) =
1 π√x(1−x) ,^0 < x <^ 1; 0 , else. By using the substitution x = sin^2 θ in the appropriate integral, find the expectation of X. [4] (ii) Let Y be a random variable with probability density function q(x) = (^) 4(1 +^1 |x|) 3 / 2 , (x ∈ R). Show that Y is not integrable. [6]
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SECTION A continued
A3. Let Ω be the set {aaa, aab,... , ccc} of all possible words that have 3 letters, such that the letters are a, b and c. Then P(E) = E/27 defines a probability measure on Ω, where E denotes the number of elements of a subset E of Ω. (i) Let Ej = {jth^ letter of chosen word is a} for j = 1, 2 , 3. Show that the events E 1 , E 2 and E 3 are mutually independent for P. [4] (ii) Now let A = {abc, acb, cab, cba, bca, bac, aaa, bbb, ccc}. Show that Q(E) = (E ∩ A)/9 is the conditional probability of E given A. [2] (iii) Are E 1 , E 2 and E 3 mutually independent for Q? Justify your answer. [4]
A4. Let W be a random variable with probability density function
p(x) =
1 − x, for 0 ≤ x < 1; 1 + x, for − 1 ≤ x < 0 0 , else. (i) Calculate the characteristic function ϕW of W. [6] (ii) Using (i), find the probability density function of X+Y , where X and Y are independent random variables, each uniformly distributed on the interval [− 12 , 12 ]. [6] (iii) Write down, but do not evaluate, a formula for the rth moment of W in terms of ϕW for each r ∈ N. [2]
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SECTION B continued
B2. Let X and Y be random variables on a probability space (Ω, P) such that 0 < E(X^2 ) < ∞ and 0 < E(Y 2 ) < ∞. (i) Show that E((X − EX)^2 ) = E(X^2 ) − (EX)^2. [4] (ii) By considering possible roots of the quadratic equation q(t) = 0 where q(t) = E(X + tY )^2 , show that (E(XY ))^2 ≤ E(X^2 )E(Y 2 ). [8] (iii) Let X have mean μ and variance σ^2. Prove Chebyshev’s Inequality P(|X − μ| > aσ) ≤ 1 /a^2 , (a > 0). [10] (iv) Suppose that Yn (n = 1, 2 ,... ) are random variables on (Ω, P) such that E(Y (^) n^2 ) < ∞ for all n and Yn converges to Y in mean square as n → ∞. Prove that Yn converges to Y in probability as n → ∞. [8]
B3. Let Z be a standard normal random variable with probability density function p(z) = √^12 π e−z^2 /^2 , (z ∈ R). (i) Calculate the probability density function of X, where X = Z^2. [6] (ii) Let X 1 , X 2 , X 3 , X 4 be mutually independent copies of X. By considering characteristic functions, show that the probability density of V = X 1 + X 2 + X 3 + X 4 is equal to
q(x) =
x 4 e−x/ (^2) , x > 0; 0 , x ≤ 0. [14] (iii) Calculate the probability density function of W = √V. [6] (iv) Determine the value of x that makes q(x) largest. [4]
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