Statistics 2024 Paper I 50 marks Calculate

Q2

(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)

हिंदी में प्रश्न पढ़ें

(a) मान लीजिए कि दो यादृच्छिक चरों X तथा Y का संयुक्त प्रायिकता घनत्व फलन f(x, y) = x/3, 0 < 2x < 3y < 6; 0, अन्यथा है। निम्नलिखित की गणना कीजिए : (i) E(Y|X = x) (10) (ii) E(var(Y|X = x)) (10) (b) यादृच्छिक चर X का बंटन फलन प्राप्त कीजिए जिसका अभिलक्षण फलन φ(t) = e^(-t²), -∞ < t < ∞ है। इसके अलावा Φ(z) के संदर्भ में P(X > 2√2) की गणना भी कीजिए, जहाँ Φ(z) = ∫_{-∞}^{z} (1/√(2π)) e^(-θ²/2) dθ. (15) (c) मान लीजिए कि X₁, X₂, ..., X₂ₙ स्वतंत्र और सर्वसम बंति (iid) N(0, 1) विचर हैं। [(X₁/X₂) + (X₃/X₄) + ... + (X₂ₙ₋₁/X₂ₙ)] / [X₁² + X₂² + ... + Xₙ²] का सीमान्त बंटन ज्ञात कीजिए। (15)

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How this answer will be evaluated

Approach

Calculate the required quantities systematically across all parts: spend approximately 40% time on part (a) covering conditional expectation and conditional variance with proper region identification; 30% on part (b) for characteristic function inversion and normal probability computation; and 30% on part (c) for establishing the limiting distribution using Slutsky's theorem and properties of ratio distributions. Begin with clear region sketches for (a), apply Fourier inversion for (b), and justify convergence arguments for (c).

Key points expected

  • For (a)(i): Correctly identify the region 0 < 2x < 3y < 6, derive marginal f_X(x) = x(2-x)/3 for 0 < x < 3, and obtain conditional pdf f_{Y|X}(y|x) leading to E(Y|X=x) = (2+x)/3
  • For (a)(ii): Compute Var(Y|X=x) = (2-x)²/36 and then calculate E[Var(Y|X)] by integrating against f_X(x) to obtain 1/30
  • For (b): Recognize φ(t) = e^{-t²} corresponds to N(0, 2) via inversion formula or matching with standard normal CF, then express P(X > 2√2) = 1 - Φ(2) using variance σ² = 2
  • For (c): Identify that numerator S_n = Σ(X_{2k-1}/X_{2k}) has E(S_n) undefined but apply Cauchy distribution properties, while denominator T_n = ΣX_i² ~ χ²_n, then use Slutsky/continuous mapping to show limiting distribution is standard Cauchy
  • Proper handling of ratio of independent quantities in (c): Show numerator terms are iid Cauchy(0,1) and denominator is χ²_n, establishing that the ratio converges to Cauchy(0,1) or requires normalization clarification

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly identifies integration regions for (a) with proper inequalities, recognizes φ(t)=e^{-t²} implies variance 2 not 1 for (b), and correctly identifies the structure of paired ratios and chi-square denominator for (c)Identifies most regions correctly but makes minor errors in bounds or misidentifies distribution parameters in (b) or (c)Fundamental errors in region identification, confuses characteristic function with standard normal, or misidentifies the limiting structure in (c)
Method choice20%10Uses conditional expectation formula E(Y|X=x) = ∫y·f_{Y|X}dy for (a), applies Fourier inversion or CF-matching for (b), and employs Slutsky's theorem/continuous mapping theorem appropriately for (c)Uses correct general methods but with some inefficient steps or misses optimal approaches like recognizing standard formsUses incorrect methods such as direct integration without conditioning for (a), attempts density convolution for (b), or ignores dependence structure in (c)
Computation accuracy20%10Accurate integration yielding E(Y|X=x)=(2+x)/3, E[Var(Y|X)]=1/30, correct N(0,2) identification with P(X>2√2)=1-Φ(2), and proper handling of Cauchy-chi-square ratio convergenceCorrect approach with minor arithmetic errors in integration bounds or coefficients, or slight errors in probability expressionsMajor computational errors leading to wrong final expressions, incorrect integration limits, or invalid probability values outside [0,1]
Interpretation20%10Explains why conditional variance decreases with x in (a), interprets the scaling in characteristic function inversion for (b), and clearly explains why the limiting distribution emerges from ratio of 'stable' numerator to diverging denominator in (c)Provides some interpretation but misses key insights about distributional properties or limiting behaviorNo interpretation of results, fails to explain why methods work, or provides incorrect statistical interpretations
Final answer & units20%10All final answers clearly stated: conditional expectation and variance expressions for (a), distribution function F(x)=Φ(x/√2) with probability in terms of Φ for (b), and explicit limiting distribution (Cauchy or properly normalized form) for (c)Most answers present but with missing components or unclear final expressionsMissing final answers, incorrect probability statements, or failure to express results in required form (especially Φ notation for (b))

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