UPSC Statistics 2024 — Paper I

(a) Two events A and B are such that P(A) = 1/3, P(B) = 1/4 and P(A|B) + P(B|A) = 2/3. Evaluate the following: (i) P(A^c ∪ B^c) (5 marks) (ii) P(A|B^c) + P(B|A^c) (5 marks) (b) Suppose the joint probability function of two random variables X and Y is f(x, y) = (xy^(x-1))/3; x = 1, 2, 3 and 0 < y < 1. Compute the following: (i) P(X ≥ 2 and Y ≥ 1/2) (5 marks) (ii) P(X ≥ 2) (5 marks) (c) Let X₁, X₂, ... is a sequence of independent and identically distributed random variables with mean (μ) and variance (σ²) < ∞, and assume Sₙ = X₁ + X₂ + ... + Xₙ. Show that WLLN does not hold for sequence ⟨Sₙ⟩ of random variables. (10 marks) (d) Write the criterion of a good estimator. Let X₁, X₂ be iid P(λ) random variables, then show that T = X₁ + X₂ is sufficient while T = X₁ + 2X₂ is not sufficient for estimating λ. (10 marks) (e) For testing H₀: μ = 100 vs. H₁: μ ≠ 100, a random sample of size 50 is drawn from a normal population with unknown mean μ and variance 200. If α = 0.05, then obtain the critical region. (10 marks)

(a) Let the joint probability density function of two random variables X and Y be f(x, y) = x/3, for 0 < 2x < 3y < 6; 0, otherwise. Compute the following: (i) E(Y|X = x) (10 marks) (ii) E(var(Y|X = x)) (10 marks) (b) Find the distribution function of random variable X, for which the characteristic function is φ(t) = e^(-t²), -∞ < t < ∞. Also compute P(X > 2√2) in terms of Φ(z), where Φ(z) = ∫₋∞^z (1/√(2π)) e^(-θ²/2) dθ. (15 marks) (c) Let X₁, X₂, ..., X₂ₙ be iid N(0, 1) variates. Find the limiting distribution of [(X₁/X₂) + (X₃/X₄) + ... + (X₂ₙ₋₁/X₂ₙ)] / [X₁² + X₂² + ... + Xₙ²]. (15 marks)

(a) Let moment generating function of random variable X exist in the neighbourhood of zero and if E(Xⁿ) = 1/5 + (-1)ⁿ 2/5 + (2ⁿ⁺¹)/5; n = 1, 2, 3, … then find the values of the following: (i) P(|X - 0.75| ≤ 1.5) (10 marks) (ii) P(|X - μ| < σ); μ = E(X) and σ² = var(X) (10 marks) [Use √1.84 = 1.36] (b) (i) Write the importance of Cramer-Rao inequality and Rao-Blackwell theorem. (5 marks) (ii) Let X ∼ B(1, θ), then find the uniformly minimum variance unbiased estimator (UMVUE) of θ(1-θ). (10 marks) (c) Obtain the maximum likelihood estimates of α and β for a random sample from the exponential population f(x; α, β) = Ce⁻β(x-α), α ≤ x < ∞, β > 0 (15 marks)

(a) Find the most powerful test of size α(= 0·05) for testing H₀: μ = 0 vs. H₁: μ = 1, given a random sample of size 25 from N(μ, 16) population. (20 marks) (b) A lot consists of some defective items. A random sample of 25 items has 6 defective items with probability p₁ = θ and 19 non-defective items with probability p₂ = 1 – θ. Then estimate θ using the following: (i) MLE method (ii) Minimum χ²-method (iii) Modified minimum χ²-method (15 marks) (c) Differentiate between Mann-Whitney U-test and Wilcoxon sign test. The following data pertain to APGAR scores of 15 pregnant women in two care programmes A and B: Programme A : 8 7 6 2 5 8 7 3 Programme B : 9 9 7 8 10 9 6 Is there a significant difference in APGAR scores of pregnant women under the two care programmes? [Given, U₍₀.₀₅₎ = 10] (15 marks)

(a) How will you justify the usage of the principle of least squares in estimating the parameters of a linear regression model? With usual notations, for the regression model y = Xβ + ε, show that the least square estimator of β is β̂ = (X'X)⁻¹X'y (10 marks) (b) (i) If X̃ is distributed as N₃(μ̃, Σ), find the distribution of [X₁ - X₂; X₂ - X₃]. (5 marks) (ii) If X₁, X₂ and X₃ are three variables, obtain the expression for the partial correlation coefficient between X₁ and X₂ eliminating the effect of X₃, ρ₁₂·₃, in terms of simple correlation coefficients. (5 marks) (c) X₁ and X₂ are independent data sets of order (n₁ × p) and (n₂ × p) respectively from Nₚ(μ̃, Σ). Show that (n₁n₂D²)/n is distributed as T²(p, n-2), where n = n₁ + n₂, and T² and D² represent the Hotelling's T² and Mahalanobis D² respectively. (10 marks) (d) For the population U = {a, b, c, d, e}, consider the following sampling design: P({a, b, d}) = 1/6, P({a, b, e}) = 1/6, P({a, d, e}) = 1/6, P({b, c, d}) = 1/6, P({b, c, e}) = 1/6, P({c, d, e}) = 1/6. Calculate the first-order and second-order inclusion probabilities. Hence show that it is a matter of a stratified design. Identify the strata with their units. (10 marks) (e) Let the incidence matrix of a design be N = [[1, 1, 1, 0], [1, 1, 0, 1], [1, 0, 1, 1], [0, 1, 1, 1]]. Show that— (i) the design is connected balanced; (ii) its efficiency factor is E = 8/9. (6+4=10 marks)

(a) (X, Y) has bivariate normal distribution BN(μ₁, μ₂, σ₁², σ₂², ρ). (i) Show that X and Y are independent if and only if ρ = 0. (6 marks) (ii) If (X, Y) follows BN(3, 1, 16, 25, 3/5), obtain P(3 < Y < 8 | X = 7), given Φ(2) = 0.9772 and Φ(-0.25) = 0.4017, and Φ(x) represents the area under the standard normal curve from -∞ to x. (6 marks) (iii) If (X, Y) follows BN(0, 0, 1, 1, 0), what will be the distribution of Z = Y/X? (4 marks) (iv) State the multivariate extension of (i) when X̃ follows Nₚ(μ̃, Σ). (4 marks) (b) Define principal components and canonical correlation. How can one attain data reduction using principal components? If (X₁, X₂) has covariance matrix Σ = [[1, ρ], [ρ, 1]], then find the principal components. (15 marks) (c) For the simple linear regression model y = β₀ + β₁x + ε, where β₀ and β₁ are parameters and ε has zero mean and an unknown variance σ², find the estimates of β₀ and β₁ by the principle of least squares as well as the method of maximum likelihood. Examine whether they are identical. (15 marks)

(a) A very big population is divided into two strata. The allocation of units of stratified random sample of size n for the two strata under Neyman allocation are n'_1 and n'_2, and under other type of allocation are n_1 and n_2. Define r = n'_1/n'_2 and μ = n_1/(rn_2). Then prove that the efficiency of stratified random sampling with respect to stratified random sampling under Neyman allocation is given by e = μ(r+1)²/((μr + 1)(μ + r)). (20 marks) (b) A bank has 40000 clients in its computer files, divided into 4000 branches, each managing exactly 10 clients. To estimate the proportion of clients for whom the bank has granted loan, a simple random sample of 40 branches is selected. From the selected sample, for each branch i, a list of clients (A_i) having a loan is prepared; i = 1, 2, ..., 40. The data observed from the selected sample are Σ(i=1 to 40) A_i = 200 and Σ(i=1 to 40) A_i² = 1156. (i) What type of sampling is this? (3 marks) (ii) State the expression of the parameter to estimate and obtain its unbiased estimate. (6 marks) (iii) Estimate the variance of the unbiased estimator obtained in part (ii). (6 marks) (c) (i) Verify whether the following BIBD are possible: (1) v = b = 22, r = k = 7, λ = 2; (2) v = 10, b = 18, r = 9, k = 5, λ = 4. Given that the design is resolvable. (ii) Given below is the incidence matrix (N) of a block design. Find the degrees of freedom associated with the adjusted treatment sum of squares and the degrees of freedom for the error sum of squares.

(a) A 2²-factorial design was used to develop the yield of a crop. Two factors A and B were used at two levels: low (–1) and high (+1). The experiment was replicated two times with completely randomized way. The data obtained are as follows: | Factor A | Factor B | Estimated Average Effect | | – | – | | | + | – | 8 | | – | + | –5 | | + | + | 2 | The sum of squares of all the yields = 510.5 The grand total of all the yields = 50.00 (i) Analyze the data and identify the significant factors. (12 marks) (ii) Develop the regression model and predict the yield when A and B both are at low level (–1). (8 marks) [Given, F₍₁, ₄, ₀.₀₅₎ = 7.71] (b) To estimate the population mean Ȳ of a characteristic Y, a simple random sample of size 1000 was selected from a population of size 1000000 by without replacement. The population mean of an auxiliary character X is X̄ = 15. The other results are given below: s²ᵧ = 20, s²ₓ = 25, sₓᵧ = 15, x̄ = 14, ȳ = 10. (i) Estimate Ȳ using difference, ratio and regression estimators. (6 marks) (ii) Estimate the MSE of these estimators. Which estimator should we choose to estimate Ȳ? (9 marks) (c) Write down the model used in the analysis of a two-way classification with interactions, stating the assumptions. What are the hypotheses tested in this scenario? Obtain the expression for the sum of squares and complete the ANOVA. (15 marks)