UPSC Statistics 2022 — Paper I

(a) Let X and Y be independent random variables with exponential distribution having respective means 1/(λ₁) and 1/(λ₂), λ₁ > 0, λ₂ > 0. Find E [max (X, Y)]. (10 marks) (b) Using Central Limit Theorem, show that limₙ → ∞ e⁻ⁿ Σₖ₌₀ⁿ (n^k)/(k!) = 1/2 (10 marks) (c) An unbiased six-sided die is thrown twice. Let X denote the smaller of the scores obtained. Then show that the probability mass function (p.m.f.) of X is given by : p_X(x) = (13-2x)/36, x = 1, 2, ..., 6 = 0, otherwise. (10 marks) (d) Let T₁ and T₂ be two unbiased estimators of θ with Var(T₁) = Var(T₂), then show that Corr(T₁, T₂) ≥ 2e – 1, where e is the efficiency of each estimator. (10 marks) (e) An urn contains 5 marbles of which θ are white and the others black. In order to test null hypothesis H₀ : θ = 3 versus alternative hypothesis H₁ : θ = 4, two marbles are drawn at random. H₀ is rejected if both the marbles are white, otherwise H₀ is accepted. Show that probability of type I error in case of without replacement and with replacement schemes, both are less than 0·40, but power of the test under with replacement is higher than that of under without replacement scheme. (10 marks)

(a) Let a random variable X have exponential distribution with mean 1/θ, θ > 0. To test H₀ : θ = 3 against H₁ : θ = 2, construct sequential probability ratio test. Show that probability of terminating the test at the first stage when null hypothesis is true is 1 – 8/27 ((A–B)/AB), where B and A, B < A, are stopping bounds. (20 marks) (b) Each Sunday a fisherman visits one of three possible locations near his home : he goes to the sea with probability 1/2, to a river with probability 1/4, or to a lake with probability 1/4. If he goes to the sea there is an 80% chance that he will catch fish; corresponding figures for the river and the lake are 40% and 60% respectively. (i) Find the probability that, on a given Sunday, he catches fish. (ii) If, on a particular Sunday, he comes home without catching anything, determine the most likely place that he has been to. (5+10=15 marks) (c) Let X₁ < X₂ < X₃ be the order statistics from uniform population having probability density function f(x; θ) = 1/θ, 0 < x < θ. Show that 4X₁ is an unbiased estimator of θ. (15 marks)

(a) (i) How large a sample must be taken in order that the probability will be at least 0·90 that the sample mean will be within 0·4 – neighbourhood of the population mean, provided the population standard deviation is 2 ? (8 marks) (ii) Examine whether the weak law of large numbers holds for the sequence Xₖ of independent random variables defined as follows : P(Xₖ = -1 - 1/k) = 1/21 - (1 - 1/(k²))¹/2, P(Xₖ = 1 + 1/k) = 1/21 + (1 - 1/(k²))¹/2. (7 marks) (b) Theoretical probabilities in the four cells of a multinomial distribution are (2+θ)/4, (1-θ)/4, (1-θ)/4 and (θ)/4, whereas the observed frequencies are 108, 27, 30 and 8 respectively, then estimate θ by maximum likelihood method. Also, obtain the standard error of the estimate. (20 marks) (c) If X is a random variable with characteristic function φ(t) = 1-|t|, & |t| ≤ 1 0, & otherwise, then obtain the corresponding probability density function. (15 marks)

(a) Consider Poisson distribution P_θ(X = j) = (e⁻θ θ^j)/(j!) = pⱼ, j = 0, 1, 2, .... Let fⱼ be the frequency for X = j and E(fⱼ) = mⱼ = npⱼ. Discuss how you obtain minimum chi-square estimate for θ. Does minimum chi-square method necessarily yield a sufficient statistic even if it exists ? (20 marks) (b) (i) Let the joint probability density function of X and Y be f(x, y) = C . exp -(4x² + 9y² - xy), where C is a constant. Find E(X), V(X), E(Y), V(Y) and the correlation coefficient between X and Y. (10 marks) (ii) If X₁, X₂, ..., X₆ are independent random variables such that P(Xᵢ = -1) = P(Xᵢ = 1) = 1/2, i = 1, 2, ..., 6, then obtain the value of P[Σᵢ₌₁⁶ Xᵢ = 4]. (5 marks) (c) The following data present the time (in minutes), that a commuter had to wait to catch a bus to reach his destination : Use the sign-test at 0·05 level of significance to test the claim of the bus operators that commuters do not have to wait for more than 15 minutes before the bus is made available to them. [Given Z₍₀.₀₂₅₎ = 1·96, Z₍₀.₀₅₎ = 1·645] (15 marks)

(a) Define general linear model with usual assumptions. If y₁ = β₁ + u₁, y₂ = –β₁ + β₂ + u₂, y₃ = –β₂ + u₃, where u₁, u₂, u₃ are mutually independent random variables with mean zero and variance σ², then find the least square estimators of β₁ and β₂. (10 marks) (b) Given X ~ N₃(μ, Σ), where μ = (2, 4, 3)' and Σ = ⎛8 2 3⎞ ⎜2 4 1⎟ ⎝3 1 3⎠ (i) find the regression function of X₁ on X₂ and X₃, and (ii) compute the conditional variance of X₁ given X₂ and X₃. (10 marks) (c) What is a uniformity trial ? Explain how it can be used to determine optimum shape and size. (10 marks) (d) In a 2⁶ – factorial experiment, the key block is given as : (1), ab, cd, ef, ace, abef, abcd, bce, cdef, acf, ade, abcdef, bde, bcf, adf, bdf. Identify the confounded effects. (10 marks) (e) If the coefficients of variation of x and y are equal and the correlation coefficient between x and y is ρ = 2/3, compute the efficiency of ratio estimator relative to the mean of a simple random sample. (10 marks)

(a) In a set of two-way classified data according to k levels of factor A and r levels of factor B, there is one observation in each cell. Show that the total number of error contrasts is (r – 1) (k – 1). (15 marks) (b) Describe with examples the technique of two-stage sampling. Obtain the variance of the sample mean under two-stage sampling without replacement. Hence, deduce the variance of the sample mean under : (i) Stratified random sampling, and (ii) Cluster sampling (20 marks) (c) (i) If X₁ = Y₁ + Y₂, X₂ = Y₂ + Y₃, X₃ = Y₃ + Y₁, where Y₁, Y₂ and Y₃ are uncorrelated random variables and each of which has zero mean and unit standard deviation, find the multiple correlation coefficient between X₃ and X₁, X₂. (ii) Let X be a 3-dimensional random vector with dispersion matrix Σ = ⎛9 3 3⎞ ⎜3 9 3⎟ ⎝3 3 9⎠. Determine the first principal component and the proportion of the total variability that it explains. (7+8=15 marks)

(a) Consider the following data given for a BIBD with v = b = 4, r = k = 3, λ = 2 and N = 12 : Analyse the design. [Given that : F₃,₅ (0·05) = 5·41] 15 (b) (i) The data matrix of a random sample of size n = 3 from a bivariate normal population BVN (μ₁, μ₂, σ₁², σ₂², ρ) is X = [6 10; 10 6; 8 2]. Test the null hypothesis H₀ : μ = μ₀ against H₁ : μ ≠ μ₀, where μ₀' = (8, 5), at 10% level of significance. [You are given : F₀.₁₀; ₂, ₁ = 49·5, F₀.₁₀; ₁, ₂ = 8·53] (ii) Suppose n₁ = 11 and n₂ = 12, observations are made on two random vectors X₁ and X₂ which are assumed to have bivariate normal distribution with a common covariance matrix Σ, but possibly different mean vectors μ₁ and μ₂. The sample mean vectors and pooled covariance matrix are X̄₁ = (-1, -1)', X̄₂ = (2, 1)', S_pooled = (7 -1; -1 5). Obtain Mahalanobis sample distance D² and Fisher's linear discriminant function. Assign the observation X₀ = (0, 1)' to either population Π₁ or Π₂. 10+10=20 (c) A sample of size n is drawn with equal probability and without replacement from a population with size N. Let Ŷ_N = Σᵣ₌₁ⁿ aᵣ yᵣ be any linear estimate of the population mean Ȳ_N, where aᵣ are constants and yᵣ denotes the value of the unit included in the sample at the rᵗʰ draw. (i) Show that Ŷ_N is an unbiased estimate of Ȳ_N if and only if Σᵣ₌₁ⁿ aᵣ = 1 (ii) Under above condition V(Ŷ_N) = (S²/N)[NΣᵣ₌₁ⁿ aᵣ² - 1] (iii) If aᵣ = 1/n, for what value of n may this variance of the sample mean in simple random sampling without replacement be exactly half the variance of the mean of a random sample of the same size taken with replacement ? 15

(a) (i) What are orthogonal polynomials ? How do you fit an orthogonal polynomial of degree 'p' ? (ii) For the model Y_(n×1) = X_(n×k) β_(k×1) + u_(n×1), E(uu') = σ² I_n, where X_(n×k) is a matrix of rank k (k < n), find out the value of E[Y'(I_n - X(X'X)⁻¹X')Y]. 10+10=20 (b) Consider an artificial population of three farms. Their selection probabilities and the wheat production (in '000 tons) are as follows : Farm unit (i) : 1 2 3; Selection probability (pᵢ) : 0·3 0·2 0·5; Wheat production (yᵢ) : 11 6 25. Draw all possible samples of size 2 with replacement (order is to be considered). Show that Horvitz-Thompson estimator of total wheat production is unbiased. 15 (c) What is a missing plot technique ? Derive the missing value formula for a Latin Square Design. How would you proceed to analyse such a design ? 15