UPSC Statistics 2023 — Paper I

Solve each sub-part systematically with clear mathematical working. For (a), apply conditional probability using survival data; (b) integrate piecewise to find CDF and evaluate tail probability; (c) apply Khinchin's WLLN condition checking variance behavior; (d) count runs and apply normal approximation for hypothesis testing; (e) state Bayes theorem and minimize posterior expected loss. Allocate approximately 2 minutes per mark, presenting each solution with clear labeling and logical flow from given information to final answer.

• (a) Correct application of chain rule for conditional probability: P(age 51) = P(survive to 15) × P(survive to 50 | 15) × P(survive 50-51 | 50) = 0.9 × 0.95 × 0.96
• (b) Proper piecewise integration of f(x) to obtain F(x) with continuity checks at x=1 and x=2, then P(X > 3/2) = 1 - F(3/2) = 5/8
• (c) Derivation that E(Xₙ) = 0, Var(Xₙ) = n^(2α), and application of Khinchin's theorem requiring (1/n²)ΣVar(Xᵢ) → 0, yielding condition α < 1/2
• (d) Correct counting of runs (r=12), expected runs μᵣ = 2n₁n₂/(n₁+n₂) + 1, variance σᵣ², and Z-test showing |Z| < 1.96 so randomness not rejected
• (e) Clear distinction: prior π(θ) represents pre-sample belief, posterior π(θ|x) ∝ L(x|θ)π(θ) updates belief; Bayes estimator under squared error loss is posterior mean E[θ|x]

Derive the joint and marginal distributions systematically using transformation of variables for part (a), spending approximately 40% of time on this highest-weighted section. For (b), construct a counterexample using independent events with probabilities decaying appropriately (e.g., P(Aₙ) = 1/n). For (c), prove the equivalence of convergence in distribution and probability for degenerate limits using the definition of degenerate distribution and properties of weak convergence. Structure: direct derivations for (a)(i)-(iii), counterexample construction with verification for (b), and rigorous proof with necessary lemmas for (c).

• For (a)(i): Apply Jacobian transformation from (X,Y,Z) to (W₁,W₂,W₃), correctly computing the Jacobian determinant and establishing support 0 < w₃ < w₂ < 1, w₁ > 0
• For (a)(ii): Integrate appropriately to obtain Gamma(3,1) for W₁, Beta(2,1) for W₂, and Beta(1,1)=Uniform(0,1) for W₃
• For (a)(iii): Verify factorization of joint PDF and conclude mutual independence of W₁, W₂, W₃ with proper statistical interpretation
• For (b): Construct valid counterexample where P(Aₙ) = 1/n (or similar), verify lim sup Aₙ = ∅ by Borel-Cantelli, yet ΣP(Aₙ) diverges
• For (c): Prove that for degenerate Y = c, Yₙ →ᵈ Y implies P(Yₙ ≤ y) → 0 or 1 appropriately, hence Yₙ →ᵖ Y using definition of convergence in probability
• For (c): Establish the converse is trivial, giving complete equivalence for degenerate limits

Prove the limit result in (a)(i) using Markov/Chebyshev inequalities; solve (a)(ii) using Bayes' theorem with clear event definitions; for (b), verify Lindeberg condition or Lyapunov's theorem to establish CLT validity; for (c), define SPRT components then derive boundaries A, B and continuation region. Allocate ~20% time to (a)(i), ~15% to (a)(ii), ~30% to (b), and ~35% to (c) given mark distribution. Structure: state definitions → apply methods → derive results → interpret findings.

• (a)(i): Application of Markov's inequality or direct variance bound to show n²P{|X|>n} ≤ E[X²I(|X|>n)] → 0
• (a)(ii): Bayes' theorem setup with events K (knows), G (guesses), C (correct); calculation of P(G|C) = P(C|G)P(G)/P(C)
• (b): Verification of Lindeberg condition or checking variance of sum → ∞; showing standardized sum converges to N(0,1)
• (b): Explicit computation that Var(Sₙ) = Σn^(2λ) → ∞ and Lyapunov condition holds for λ < 1/2
• (c): Definition of SPRT, OC function L(θ), and ASN Eθ(N); derivation of Wald's boundaries A ≈ (1-β)/α, B ≈ β/(1-α)
• (c): Specific SPRT construction for N(θ,1): continuation region as sum(Xi - 4.5) between adjusted log-boundaries

Explain the theoretical foundations and derive the required estimators and tests across all three parts. Spend approximately 40% of effort on part (a) covering completeness, sufficiency and UMVUE derivation; 30% on part (b) for chi-square goodness-of-fit test with correct degrees of freedom and conclusion; and 30% on part (c) for likelihood ratio test definition and derivation in normal distribution with unknown variance. Structure with clear headings for each sub-part, stating definitions first, then derivations, and ending with explicit final answers.

• Part (a): Define completeness and sufficiency; explain their roles in reducing data without loss of information and enabling unbiased estimation via Rao-Blackwell and Lehmann-Scheffé theorems
• Part (a): For U(0,θ), identify sufficient statistic T = X_(n), prove completeness, and derive UMVUE of θ as ((n+1)/n)X_(n) with proper justification
• Part (b): Set up H₀: p = 0.5 (equal probability), calculate expected frequencies under Binomial(4, 0.5), compute chi-square statistic correctly
• Part (b): Use correct degrees of freedom = 4 (not 5), compare with critical value 9.488, and state conclusion about hypothesis
• Part (c): Define Likelihood Ratio Test (LRT) as λ(x) = sup_{θ∈Θ₀}L(θ)/sup_{θ∈Θ}L(θ)
• Part (c): Derive LRT statistic for N(θ,σ²) with unknown σ², showing it reduces to t-test with rejection region |t| > t_{α/2,n-1}

Solve this multi-part numerical problem by allocating time proportionally to marks: spend ~20% on (a)(i)-(ii) combined, ~20% on (b), ~20% on (c), ~20% on (d), and ~20% on (e). Begin each sub-part by stating the relevant theorem or formula, show complete derivation/calculation steps, and conclude with precise final answers. For (d), complete the ANOVA table systematically before hypothesis testing. For (e), clearly define the estimator before deriving bias and MSE.

• (a)(i) Apply linear transformation theorem: if Y = AX, then Y ~ N₂(Aμ, AΣA') with correct matrix A = [[1,-1,0],[0,1,-1]]
• (a)(ii) Use independence condition Cov(X₁+X₂+X₃, X₁-X₂-X₃) = 0; solve for ρ = -1/2 and verify validity
• (b) Prove both directions: (⇒) by definition of MVN, (⇒) using characteristic functions or Cramér-Wold theorem
• (c) Set up F-test for H₀: β₀=β₁=0 using extra sum of squares; compute F = [(SSR/2)]/[SSE/(n-2)] with correct df
• (d) Complete ANOVA table: Blocks df=5, Treatments df=3, Total df=23, Total SS=33, Error MS=0.8; compute F_Treatments=6.25 and compare with critical value
• (e) Define regression estimator Ŷ_reg = ȳ + b(X̄ - x̄); derive bias ≈ 0 and MSE ≈ (1-f)S²_y(1-ρ²)/n to first order

Solve this multi-part numerical and theoretical question by allocating approximately 35% time to part (a) due to its 15 marks and computational complexity, 25% to part (b) covering ANOVA and estimability, and 40% to part (c) on PCA and canonical correlations. Begin with clear problem identification for each sub-part, show all computational steps with matrix operations for (a), present structured ANOVA decomposition for (b)(i) and rigorous linear algebra proof for (b)(ii), and provide conceptual clarity with real-world Indian examples for (c). Conclude each part with precise final answers and interpretations.

• Part (a)(i): Correctly partition Σ into Σ₁₁, Σ₁₂, Σ₂₁, Σ₂₂ and apply conditional distribution formula N₂(μ₁ + Σ₁₂Σ₂₂⁻¹(x₃-μ₃), Σ₁₁ - Σ₁₂Σ₂₂⁻¹Σ₂₁) with x₃=2
• Part (a)(ii): Compute partial correlation ρ₁₂.₃ = (σ₁₂ - σ₁₃σ₂₃/σ₃₃)/√[(σ₁₁-σ₁₃²/σ₃₃)(σ₂₂-σ₂₃²/σ₃₃)] and multiple correlation R₁.₂₃ = √[σ₁'Σ₂₂⁻¹σ₁/σ₁₁] where σ₁' = (σ₁₂, σ₁₃)
• Part (b)(i): Describe two-way ANOVA with replication: model yᵢⱼₖ = μ + αᵢ + βⱼ + (αβ)ᵢⱼ + εᵢⱼₖ, assumptions (normality, homoscedasticity, independence), ANOVA table with SS_T, SS_A, SS_B, SS_AB, SS_E, and examples like agricultural field trials (ICRISAT crop studies) or industrial quality control
• Part (b)(ii): Set up design matrix X, show rank deficiency, derive condition for estimability via Cβ where C = (p₁,p₂,p₃), prove p₁ = p₂ + p₃ using linear independence of rows and estimability condition C = LX for some L
• Part (c)(i): Three reasons for PCA: dimensionality reduction (e.g., reducing NSSO household survey variables), multicollinearity remediation in regression, and data visualization/pattern detection in large datasets
• Part (c)(ii): Define canonical correlations as correlations between linear combinations u=a'X and v=b'Y maximizing correlation; examples: relationship between economic indicators and social development indices, or agricultural inputs vs outputs; describe eigenvalue solution of Σ₁₁⁻¹Σ₁₂Σ₂₂⁻¹Σ₂₁ and extraction of canonical variates

Begin with a clear conceptual distinction in part (a) between variables (quantitative) and attributes (qualitative) with Indian examples like agricultural yield vs literacy status. Derive the unbiased estimator p̂ = n'/n for population proportion and its variance V(p̂) = (N-n)/(N-1) · p(1-p)/n, then obtain unbiased estimator v(p̂). For part (b), verify Neyman optimum allocation by checking if nᵢ ∝ NᵢSᵢ/√cᵢ (assuming equal costs), then compute V(ȳ_st). For part (c), contrast SRS with PPS on selection probability basis, then detail both Cumulative Total and Lahri's methods with numerical illustration. Allocate approximately 40% time to part (a), 30% each to (b) and (c) based on marks distribution.

• Part (a): Clear distinction between sampling for variables (measurable quantities like income, yield) vs attributes (dichotomous characteristics like employment status, disease presence) with appropriate Indian examples
• Part (a): Derivation of unbiased estimator p̂ = n'/n for population proportion P, its variance V(p̂) = (N-n)/(N-1) · P(1-P)/n under SRSWOR, and unbiased estimator of variance v(p̂) = (N-n)/(N-1) · p̂(1-p̂)/(n-1)
• Part (b): Verification of Neyman optimum allocation condition nᵢ/n = NᵢSᵢ/ΣNⱼSⱼ using given data; calculation showing existing allocation matches or approximates this ratio
• Part (b): Computation of stratified mean estimate ȳ_st = ΣWᵢȳᵢ where Wᵢ = Nᵢ/N, and population variance estimate V(ȳ_st) = ΣWᵢ²(Nᵢ-nᵢ)/(Nᵢnᵢ) · sᵢ²
• Part (c): Systematic comparison of SRS (equal probability) vs PPS (probability ∝ size) on grounds of efficiency, especially for skewed populations like industrial output or agricultural holdings
• Part (c): Step-wise description of Cumulative Total Method: list cumulative totals, select random numbers between 1 and ΣXᵢ, identify selected units
• Part (c): Step-wise description of Lahri's Method: select random number i from 1 to N and random number j from 1 to M (M=max size), accept if j ≤ Xᵢ, else reject and repeat

Begin with a structured comparison of RBD vs BIBD in part (a), then rigorously prove all three BIBD parameters using standard notation with clear algebraic steps. For part (b), first define confounding with factorial design context, then explicitly construct the three replication schemes showing which treatment combinations go to which block. For part (c), use tabular comparison for model types with agricultural/industrial examples, then explain how RBD satisfies randomization, replication, and local control. Allocate approximately 40% time to part (a) given its 20 marks and proof demands, 30% each to parts (b) and (c).

• Part (a): Clear distinction between RBD (complete blocks, all treatments per block) and BIBD (incomplete blocks, not all treatments appear in each block) with structural conditions
• Part (a): Correct proofs of bk = vr, λ(v–1) = r(k–1), and b ≥ v using incidence matrix properties or combinatorial counting with λ defined as pairwise concurrence
• Part (b): Accurate definition of confounding as sacrificing higher-order interaction information to achieve block homogeneity, with distinction between complete and partial confounding
• Part (b): Correct construction of three replications: Rep I confounds ABC (assign +++ and +–– to Block 1, ++– and +–+ to Block 2, etc.), Rep II confounds AC, Rep III confounds BC using Yates notation
• Part (c): Precise differentiation of fixed (levels specifically chosen, inference only to those levels), random (levels random sample from population, variance component estimation), and mixed models with appropriate examples like crop varieties vs fertilizer doses
• Part (c): Explanation of how RBD achieves randomization (random allocation within blocks), replication (multiple blocks), and local control (homogeneous blocks reducing experimental error)