UPSC Statistics 2026

(a) (i) Show that for n events A₁, A₂, ⋯, Aₙ P( ⋂_{i=1}^{n} Aᵢ ) ≥ Σ_{i=1}^{n} P(Aᵢ) − (n − 1) (7 marks) (ii) For two mutually exclusive events A and B, show that P(A | A ∪ B) = P…

(a) (i) Find the moment generating function (m.g.f.) of a random variable X with probability density function (p.d.f.) f(x) = { x , 0 ≤ x < 1 2 - x, 1 ≤ x < 2 0 , otherwise Using…

(a) Let the joint probability density function of X and Y be f(x, y) = 3/4 ( xy + x²/2 ) ; 0 < x < 1, 0 < y < 2 Find— (i) the marginal probability density functions of X and Y; (i…

(a) Let X₁, X₂, ⋯, Xₙ be i.i.d. random variables following N(μ, σ²). Find the maximum likelihood estimator for σ² and examine whether the same is unbiased. If biased, suggest an u…

(a) Let Y₁, Y₂, ⋯, Yₙ be n independent observations from a population with mean μ and variance σ². Obtain the best linear unbiased estimator of μ and an unbiased quadratic estimat…

(a) Let l′β be any estimable linear function of parameters β₁, β₂, ⋯, β_p. Then prove that— (i) there exists a unique linear function c′Y of random variables Y₁, Y₂, ⋯, Y_n such t…

The following data are from a two-factor random effects model : | Factor B \ Factor A | B₁ | B₂ | B₃ | | A₁ | 6, 8 | 10, 12 | 14, 16 | | A₂ | 12, 14 | 16, 18 | 20, 22 | (a) (i) Gi…

(a) Define multivariate normal distribution and write down its important properties. Let X₁, X₂, X₃ and X₄ are mutually independent random vectors following N_p(μ, Σ). Then find t…

(a) Suppose that a given individual in a population has a survival time which is exponential with a hazard rate θ. Each individual's hazard rate θ is potentially different and is…

(a) (i) Describe the construction of a CUSUM chart with Decision interval and Reference value. (10 marks) (ii) One implementation of the CPU-cache-main memory subsystem of the PDP…

(a) How will you draw the OC curve for a single sampling plan with sample size 5, if the acceptance number is c = 2, assuming the lot size to be large ? Explain. (15 marks) (b) Ea…

(a) (i) Discuss the theoretical basis of the sequential probability ratio test (SPRT) in acceptance sampling. Explain how acceptance and rejection numbers are chosen to satisfy α…

(a) Explain the 2-stage least squares (2SLS) method of estimation for the following simultaneous equation systems : C_t = α + βY_t + u_t Y_t = C_t + Z_t where C = Aggregate consum…

(a) Define autocorrelation of order k and correlogram. For an infinite series generated by the moving average of random components with equal weights, obtain the correlogram of a…

(a) (i) Define a life table and mention its uses. (5 marks) (ii) The number of persons dying at age 75 is 476, and the complete expectations of life at ages 75 and 76 years are 3·…

(a) Consider the following structural equation model assuming Y's as endogenous and X's as pre-determined variables : Y₁ = α₁₀ + α₁₂Y₂ + α₁₃Y₃ + β₁₁X₁ + u₁ Y₂ = α₂₀ + α₂₃Y₃ + β₂₁X…