MATH c UNIVERSITY OF LEEDS Examination for the Module MATH2715 (January 2015) STATISTICAL METHODS. Time allowed: 2 hours

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1 MATH2750 This question paper consists of 8 printed pages, each of which is identified by the reference MATH275. All calculators must carry an approval sticker issued by the School of Mathematics. c UNIVERSITY OF LEEDS Examination for the Module MATH275 (January 205) STATISTICAL METHODS Time allowed: 2 hours Attempt ALL questions in Section A and no more than THREE questions from Section B. Section A and B are each worth 50% of the examination marks. Questions A to A20 carry equal marks. Questions A to A0 require you to write down a single letter answer. Questions A to A20 require you to write down a short explanation or draw a sketch. Your answers to Section A questions and Section B questions may be written in the same answer book. CONTINUED...

2 SECTION A Attempt ALL questions in Section A Questions A to A0 require you to write down a single letter answer A. A continuous random variable X has probability density function f X (x) = θ, x >, x+θ and f X (x) = 0 for x, and where θ >. If U = /X, what does the probability density function f U (u) equal (where f U (u) is non-zero)? A: θu θ+, B: θu θ, C: θu +θ, D: θ u θ. A2. In question A above, what is the range where f U (u) is non-zero? A: 0 < u <, B: 0 < u <, C: < u <, D: < u <. A3. A random variable X has moment generating function m X (t) = E[e tx ] given by m X (t) = a 0 + a t + a 2 t 2 + a 3 t 3 +. What is the value of E[X 2 ]? A: a, B: 2 a 2, C: a 2, D: 2a 2. A4. A random variable X has zero mean and variance one. What is the maximum possible value for pr{ X 2}? A: 0.05, B: 0.25, C: 0.50, D: A5. Suppose that random variables X, X 2, X 3 satisfy Var[X i ] = for i =, 2, 3 and corr(x, X 2 ) = corr(x 2, X 3 ) = ρ and corr(x, X 3 ) = 0. What does Var[X + X 2 + X 3 ] equal? A: 3 + ρ, B: 3 + 2ρ, C: 3 + 4ρ, D: 3 + 6ρ. 2 CONTINUED...

3 A6. In question A5 above, what does cov(x + X 2, X + X 3 ) equal? A: 2, B: + 2ρ, C: 2 + 4ρ, D: 4 + 3ρ. A7. Suppose random variables X, X 2,..., X n are independent and identically distributed with common mean µ and common variance σ 2 and X n = n X i. What is n n the value of (X i µ)? i= A: n( X n µ), B: 0, C: E[ X n ], D: i= j= i= n n (X i µ)(x j µ). A8. If X and Y are independent standard normal random variables, what is the distribution of U = X Y? A: N(0, 0), B: N(0, ), C: N(0, 2), D: N(0, 4). A9. Suppose that (X, Y ) have a bivariate normal distribution with variances σx 2 = 9 and σy 2 = 4, and corr(x, Y ) =. What is the variance-covariance matrix Σ of (X, Y )? 2 ( ) ( ) ( ) ( ) A: Σ =, B: Σ = 9, C: Σ = 2 9 3, D: Σ = ( ) a b A0. If M = is an orthogonal matrix, what does the inverse M equal? b a ( ) ( ) ( ) ( ) a b a b a b a b A:, B:, C:, D:. b a b a b a b a 3 CONTINUED...

4 Questions A to A20 require you to write down a short answer or draw a sketch. A. A continuous random variable X has probability density function f X (x) = where θ >. Obtain the mean E[X]. θ, x >, x+θ A2. If a random variable X has probability density function as given in question A above, obtain the cumulative distribution function F X (x) of X. A3. Suppose that X and Y have joint probability density function f XY (x, y) while f X (x) is the marginal probability density function of X. Write down an expression for f(y = y X = x), the conditional probability density function of Y given X = x. A4. If u = x y and v = y, sketch the region in the u-v plane corresponding to the region 0 < x < 2 and 0 < y < 2. A5. A random variable Y is used to estimate an unknown parameter θ. Define the mean square error of Y. A6. In question A5 above, define the bias of Y. A7. For a random variable X with probability density function f X (x), define the characteristic function φ X (t). A8. Suppose that X and X 2 are independent Cauchy random variables, each with common characteristic function φ X (t) = e t. Obtain the characteristic function of Y = 2 (X + X 2 ) and so deduce the distribution of Y. A9. Suppose a probability distribution involves a single unknown parameter θ. Briefly comment on how a classical frequentist and a Bayesian statistician might interpret the statement pr{θ < } = 0.2. A20. In Bayesian methodology, what do you understand by the term conjugate prior? (It is not necessary to explain what is meant by a prior.) 4 CONTINUED...

5 SECTION B Attempt no more than THREE questions from this section B. Random variables X and Y have joint probability density function f XY (x, y) = xe x(y+), x > 0, y > 0. Show that the marginal probability density function of Y is f Y (y) = ( + y) 2, y > 0. Derive also the marginal probability density function f X (x) of X. Are X and Y independent? Justify your answer. For the transformation U = X(Y + ) and V = Y obtain the joint probability density function f UV (u, v) of (U, V ). Obtain the marginal probability density functions of U and V, f U (u) and f V (v) respectively. Are U and V independent? Justify your answer. [ You may use the result that: 0 λt n e λt dt = ] (n )! for integer n. λ n 5 CONTINUED...

6 B2. The random variable X has an exponential distribution with parameter θ and probability density function f X (x) = θe θx, x > 0. Obtain the moment generating function m X (t) = E[e tx ] of X. For what values of t is this defined? Show that the mean and standard deviation of X both equal θ. If U = X Y where X and Y are independent exponential random variables with parameter θ, show that U has moment generating function m U (t) = θ2 θ 2 t 2. Suppose that U,...,U n are mutually independent and identically distributed random variables each having moment generating function m U (t) as above. Let S n = U + + U n. Write down, together with a brief statement of what result you have used, the moment generating function of S n. Show that Y = θs n has moment generating function 2n m Y (t) = ) n ( t2. 2n Obtain log m Y (t) and deduce what happens as n. What do you conclude about the distribution of Y for large n? You may use the results that: (i) if W N(µ, σ 2 ), then it has moment generating function m W (t) = exp(µt + 2 σ2 t 2 ), (ii) if v <, then log( v) = v v2 2 v CONTINUED...

7 B3. A continuous random variable X has an inverse-gamma(α, λ) distribution if it has probability density function ( ) f X (x) = λα λ Γ(α) x α exp, x > 0, x where α > 0 is a known constant, λ > 0 is an unknown parameter, and Γ(α) is the gamma function satisfying Γ(α) = (α )Γ(α ). (a) In the case α >, show that X has mean E[X] = λ α. (b) If x = (x, x 2,..., x n ) are a random sample from this distribution, show that the log-likelihood function is l(λ; x) = nα log(λ) n log(γ(α)) (α + ) log(x x 2...x n ) n i= λ x i. Hence, or otherwise, show that the maximum likelihood estimate of λ is λ = n nα. i= x i By differentiating l(λ; x) a second time, or otherwise, verify that λ does indeed maximise the log-likelihood. (c) For n large, obtain an expression for Var[ λ]. (d) Obtain the method of moments estimate for λ. 7 CONTINUED...

8 B4. Suppose Y gamma(α, λ) with probability density function f Y (y) = λα y α e λy, y 0, Γ(α) for α > 0 and λ > 0, where Γ(α) is the gamma function satisfying Γ(α + ) = αγ(α). Obtain the mean of Y. Show that the mode of Y equals α λ in the case where α. Suppose x = (x,..., x n ) forms a random sample from a Poisson distribution with probability function p(x µ) = µx e µ, x = 0,, 2,..., x! where µ is the unknown mean. Suppose further that the prior distribution for µ is a gamma(a, b) distribution where a and b are known positive quantities. Show that the posterior distribution satisfies µ x gamma(a + n x, b + n) where x is the sample mean of the n observations. What is the mean and mode of the posterior distribution of µ? If a large amount of data is available, so n is large, what is the mode of the posterior distribution of µ? Comment briefly on your result. Consider now the case where n is finite and the prior density of µ satisfies π(µ) for µ > 0. Determine the mode of the posterior distribution of µ x and comment briefly on your result. END 8