UPSC Statistics 2024

(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) Su…

(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…

(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 fo…

(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…

(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 =…

(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,…

(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, a…

(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 wit…

(a) Consider a system consisting of three identical units connected in parallel. The unit reliability factor is 0·90. If the unit failures are independent of one another, and if t…

(a) Obtain the control limits for X̄-chart and R-chart and describe the significance of joint study of these charts. 20 marks (b) Find the reliability and hazard functions of Weib…

(a) With respect to a given Linear Programming Problem (LPP), explain the following concepts : 15 marks (i) Extreme Point Solutions (ii) Duality Theorem (iii) Complementary Slackn…

(a) A manually handled toll-booth has two tellers, who are each capable of handling an average of 60 vehicles per hour, with the actual service times exponentially distributed. Ve…

(a) Discuss the Indian Statistical System. State some important organisations and explain the main working of the National Statistical Organisation (NSO). 10 marks (b) Consider AR…

(a) For the linear model Y = Xβ + u, obtain the expressions for Ordinary Least Squares (OLS) and Generalised Least Squares (GLS) estimators of the parameters. Discuss their proper…

(a) Explain the problem of identification with a suitable example. Also discuss the conditions of identification. Check the identifiability of each equation of the following struc…

(a) Four items are to be constructed so that they are equispaced on the difficulty scale. If the easiest item is passed by 85% of the group and the most difficult by 25%, find the…