(a)(i) If (X, Y) follows bivariate normal BN(μ₁, μ₂, σ₁², σ₂², ρ), then obtain (A) E(e^X) (B) E(e^(X+Y)) (C) Var(e^X) and (D) Correlation between e^X and e^Y. 3+3+3+3=12 marks

(a)(ii) If (X, Y) have the joint probability density function g(x,y) = y

Question

(a)(i) If (X, Y) follows bivariate normal BN(μ₁, μ₂, σ₁², σ₂², ρ), then obtain (A) E(e^X) (B) E(e^(X+Y)) (C) Var(e^X) and (D) Correlation between e^X and e^Y. 3+3+3+3=12 marks

(a)(ii) If (X, Y) have the joint probability density function g(x,y) = y e^(-y(x+1)), for x ≥ 0, y ≥ 0; 0 elsewhere, then find the regression curve of X on Y and comment on the nature of the curve. 8 marks

(b) Let X = (X₁, X₂, X₃)' ~ N₃(μ, Σ), in which μ = (2 1 3)' and Σ = (9 2 -2 / 2 2 -3 / -2 -3 9). Obtain (i) E{X₁ | X₂ = x₂, X₃ = x₃} and (ii) Var{X₁ | X₂ = x₂, X₃ = x₃}. 15 marks

(c) Consider the model: Y = X θ + ε, where ε is an n×1 vector of unobservable random variables such that E(ε) = 0 and D(ε) = σ²Ω, σ>0 unknown, Ω is a positive definite matrix of known constants and rank(X) = k<n. Then (i) Derive least square estimator of θ and (ii) Derive an unbiased estimator of σ². 9+6=15 marks

UPSC Answer Check · Accepted Answer

Derive all required quantities systematically across three parts: spend ~35% time on (a) covering MGF technique for lognormal moments and regression curve derivation; ~30% on (b) for conditional multivariate normal using Schur complement; and ~35% on (c) for GLS estimator derivation via Aitken transformation and unbiased variance estimation. Begin each part with appropriate distribution assumptions, show complete derivation steps, and conclude with explicit final expressions.
- Part (a)(i): Use MGF of bivariate normal to derive E(e^X)=exp(μ₁+σ₁²/2), E(e^(X+Y))=exp(μ₁+μ₂+(σ₁²+σ₂²+2ρσ₁σ₂)/2), Var(e^X), and Corr(e^X,e^Y) using lognormal properties
- Part (a)(ii): Obtain marginal of Y, conditional density of X|Y, derive E(X|Y=y)=1/y showing hyperbolic regression curve with negative association
- Part (b): Apply conditional multivariate normal formula with Σ₂₂ partition, compute Σ₁₂Σ₂₂⁻¹ for conditional mean and Σ₁₁-Σ₁₂Σ₂₂⁻¹Σ₂₁ for conditional variance
- Part (c)(i): Derive GLS estimator θ̂=(X'Ω⁻¹X)⁻¹X'Ω⁻¹Y via Aitken transformation or direct minimization of generalized sum of squares
- Part (c)(ii): Derive unbiased estimator σ̂²=(Y-Xθ̂)'Ω⁻¹(Y-Xθ̂)/(n-k) using trace properties and idempotent matrix arguments
- Correct handling of positive definiteness conditions for Ω and invertibility requirements throughout
- Proper verification that E(θ̂)=θ (unbiasedness) and E(σ̂²)=σ² in part (c)

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	18%	9	Correctly identifies bivariate normal MGF, proper joint density support in (a)(ii), accurate Σ partitioning in (b) with correct submatrix identification, and proper specification of GLS assumptions including rank conditions in (c)	Basic distribution assumptions stated but with minor errors in support specification or Σ indexing; GLS setup mostly correct but missing explicit rank verification	Confuses conditional with marginal distributions, wrong density support, or fundamentally misidentifies the transformation needed for GLS
Method choice	22%	11	Selects MGF approach for (a)(i) moments, integration by parts or recognition of Gamma/Exponential forms for (a)(ii), Schur complement formula for (b), and Aitken transformation or Lagrangian for (c); justifies why each method is optimal	Uses correct general methods but with suboptimal choices (e.g., direct integration instead of MGF properties); applies formulas without showing why they apply	Attempts inappropriate methods like naive OLS for GLS or ignores conditional structure in multivariate normal; uses moment generating without recognizing lognormal connection
Computation accuracy	24%	12	Flawless execution: correct MGF exponents, accurate 2×2 and 3×3 matrix inversions in (b), precise Σ₂₂⁻¹ computation, correct (X'Ω⁻¹X)⁻¹ derivation, and exact unbiasedness verification with proper degrees of freedom (n-k)	Minor arithmetic slips in exponent algebra or matrix elements; correct structure but computational errors in final numerical coefficients; off-by-one errors in degrees of freedom	Major computational errors: wrong determinant, incorrect matrix inversion, confused σ₁² with σ₁, or fundamental errors in quadratic form expectations
Interpretation	18%	9	Interprets regression curve E(X\|Y=y)=1/y as rectangular hyperbola showing inverse relationship; explains geometric decay of correlation in lognormal transformation; discusses efficiency gain of GLS over OLS via Gauss-Markov extension	States curve is decreasing but misses hyperbolic classification; notes GLS is 'better' without explaining BLUE property in transformed model	No interpretation of regression curve shape; treats derived formulas as endpoints without connecting to statistical meaning or practical implications
Final answer & units	18%	9	All 12 quantities explicitly boxed: four in (a)(i), regression curve with domain in (a)(ii), conditional mean and variance expressions in (b), and final matrix-form estimators in (c); dimensions clearly stated for matrix results	Final answers present but some buried in text; missing explicit statement of conditional variance formula or unclear on estimator dimensions	Missing final answers for sub-parts, incorrect dimensional consistency (e.g., scalar where matrix required), or answers without proper mathematical closure

Q6

Directive word: Derive

How this answer will be evaluated

Approach

Key points expected

Evaluation rubric

Practice this exact question

More from Statistics 2025 Paper I