Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
BSc and MSci EXAMINATIONS (MATHEMATICS) May-June 2010 This paper is also taken for the relevant examination for the Associateship. M2S1 PROBABILITY AND STATISTICS II Date: Monday, 17 May, 2010 Time: 2 pm – 4 pm Credit will be given for all questions attempted but extra credit will be given for complete or nearly complete answers. Calculators may not be used. Formula sheets are included as pages 4 and 5. c 2010 Imperial College London M2S1 Page 1 of 5 1. (a) Explain what is meant by a σ−field A of subsets of a sample space Ω. Let A and B belong to some σ−field A. Show that A contains the set A ∩ B. What is meant by a probability space? (b) Let X and Y have joint probability density of the form fX,Y (x, y) = cx2 y if x2 ≤ y ≤ 1, 0 otherwise. Find the value of c and P (X ≥ Y ). (c) What does it mean to say that a sequence of random variables X1 , X2 , . . . , (i) converges in probability, (ii) converges in distribution to a random variable X? (d) For each of the following statements state, without proof, whether it is True or False. (i) Maximum likelihood estimators are always unbiased. (ii) In testing of a hypothesis H0 , the p−value of the test is the probability that H0 is true. (iii) Convergence in distribution always implies convergence in probability. (iv) If a sequence of random variables X1 , X2 , . . . converges in probability to the constant b, then E[Xn ] → b. 2. (a) Consider a random variable X with the Laplace distribution, so that the probability density function of X is fX (x) = 1 exp(−|x|), −∞ < x < ∞. 2 Derive the moment generating function, MX (t), of X and state the interval around 0 that t must be in for MX (t) to be well-defined. By considering the moment generating function or otherwise, evaluate the expectation and variance of X. (b) Now consider independent random variables X and Y both of which have the Laplace distribution described in part (a). Let U = X + Y, V = X − Y . Derive the moment generating function of U and show that U and V are identically distributed. Find the joint moment generating function of U and V . Show that U and V are uncorrelated but not independent. M2S1 PROBABILITY AND STATISTICS II (2010) Page 2 of 5 3. (a) Let X and Y be independent non-negative random variables, with densities fX and fY respectively. Find the joint density of U = X and V = X + aY , where a is a positive constant. Let X and Y be independent and exponentially distributed random variables, each with density f (x) = λ exp(−λx), x ≥ 0. Find the density of X + 12 Y . Is it the same as the density of the random variable max{X, Y }? (b) Show that if Y is a random variable with moment generating function MY (t), then for γ > 0 and t > 0, MY (t) P (Y ≥ γ) ≤ . eγt [Hint: recall Markov’s Inequality]. Hence show that if Z has a standard normal distribution, then for γ ≥ 0, P (Z ≥ γ) ≤ e−γ 2 /2 . 4. (a) Let X1 , . . . , Xn be independent, identically distributed N (μ, σ 2 ), where both μ and σ 2 are unknown. What is the distribution of Pn i=1 (Xi − μ)2 /σ 2 ? ˉ = n−1 Pn Xi State, without proof, the joint distribution of the random variables X i=1 P ˉ 2. and (n − 1)S 2 /σ 2 , where S 2 = (n − 1)−1 ni=1 (Xi − X) Explain clearly how the joint distribution allows construction of an appropriate test statistic for testing the null hypothesis H0 : μ = μ0 against the alternative hypothesis H1 : μ 6= μ0 . Describe in detail how you would carry out the test. (b) Let Y1 , . . . , Yn be independent, identically distributed N (0, σ 2 ) random variables. Pn 2 Stating, without proof, the distribution of i=1 Yi , show that the probability density pP n 2 function of W = i=1 Yi is of the form: fW (w) = (c) 2 (2σ 2 )n/2 Γ(n/2) wn−1 exp{−w2 /(2σ 2 )}. Let W1 , . . . , Wp be independent, identically distributed, with the probability density fW (w) derived in part (b). Find the maximum likelihood estimator of σ, based on W1 , . . . , Wp . M2S1 PROBABILITY AND STATISTICS II (2010) Page 3 of 5 M2S1 PROBABILITY AND STATISTICS II (2010) Page 4 of 5 n∈ λ ∈ R+ θ ∈ (0, 1) n∈ n∈ {0, 1, ..., n} {0, 1, 2, ...} {1, 2, ...} {n, n + 1, ...} {0, 1, 2, ...} Binomial(n, θ) P oisson(λ) Geometric(θ) N egBinomial(n, θ) or ∈ (0, 1) Z+ , θ n+x−1 n θ (1 − θ)x x x−1 n θ (1 − θ)x−n n−1 and the LOCATION/SCALE transformation Y = μ + σX gives y−μ 1 y−μ FY (y) = FX fY (y) = fX σ σ σ MY (t) = eμt MX (σt) 0 EfY [Y ] = μ+σEfX [X] n(1 − θ) θ n θ (1 − θ) θ2 1 − (1 − θ)x 1 θ (1 − θ)x−1 θ λ λ e−λ λx x! n θ t 1 − e (1 − θ) θet 1 − et (1 − θ) θet 1 − et (1 − θ) n n exp λ et − 1 1 − θ + θet 1 − θ + θet MX MGF VarfY [Y ] = σ 2 VarfX [X] n(1 − θ) θ2 n(1 − θ) θ2 nθ(1 − θ) nθ θ(1 − θ) VarfX [X] n x θ (1 − θ)n−x x EfX [X] θ FX CDF θx (1 − θ)1−x MASS FUNCTION fX For CONTINUOUS distributions (see over), define the GAMMA FUNCTION Z ∞ Γ(α) = xα−1 e−x dx ∈ (0, 1) ∈ (0, 1) Z+ , θ Z+ , θ {0, 1} θ ∈ (0, 1) PARAMETERS Bernoulli(θ) X RANGE DISCRETE DISTRIBUTIONS M2S1 PROBABILITY AND STATISTICS II (2010) Page 5 of 5 α, β ∈ α, β ∈ R+ R+ R+ θ, α ∈ α, β ∈ R+ R+ (0, 1) P areto(θ, α) Beta(α, β) R+ ν ∈ R+ R Student(ν) (standard model μ = 0, σ = 1) μ ∈ R, σ ∈ R+ R+ λ∈ R+ α<β∈R R+ (α, β) R N ormal(μ, σ 2 ) (standard model β = 1) W eibull(α, β) (standard model β = 1) Gamma(α, β) (standard model λ = 1) Exponential(λ) (standard model α = 0, β = 1) U nif orm(α, β) X PARAMS. (x − μ)2 exp − 2σ 2 2πσ 2 1 α Γ(α + β) α−1 x (1 − x)β−1 Γ(α)Γ(β) αθ α (θ + x)α+1 − 12 ν+1 (πν) Γ 2 (ν+1)/2 ν x2 Γ 1+ ν 2 √ αβxα−1 e−βx β α α−1 −βx e x Γ(α) λe−λx 1 β−α fX 1− α θ θ+x 1 − e−βx 1− e−λx FX x−α β−α α CONTINUOUS DISTRIBUTIONS PDF CDF (if ν > 1) α α+β θ α−1 (if α > 1) 0 μ Γ (1 + 1/α) β 1/α 2 VarfX [X] (if ν > 2) (α + β)2 (α αβ + β + 1) αθ2 (α − 1)(α − 2) (if α > 2) ν ν−2 σ2 Γ (1 + 2/α) − Γ (1 + 1/α)2 β 2/α α β2 1 λ2 1 λ α β (β − α) 12 (α + β) 2 EfX [X] α 2 t2 /2} β β−t e{μt+σ λ λ−t MX − eαt t (β − α) eβt MGF