Statistics 2023 Paper I 50 marks Derive

Q2

(a) Let X, Y, Z be three mutually independent standard exponential variates and W₁ = X + Y + Z, W₂ = (X + Y)/(X + Y + Z), W₃ = X/(X + Y). Then (i) determine the joint distribution of W₁, W₂ and W₃. (ii) find out the marginal probability density functions of W₁, W₂ and W₃. (iii) examine the mutual independence of W₁, W₂ and W₃, and give your comment. (10+6+4=20 marks) (b) Give an example to prove or disprove the following : P(lim sup Aₙ) = 0 ⇒ Σₖ₌₁^∞ P(Aₖ) < ∞, for any sequence {Aₙ, n ≥ 1} of events defined on a probability space (Ω, 𝓐, P). (15 marks) (c) Let {Yₙ, n ≥ 1} be a sequence of random variables and Y be a degenerate random variable. Examine whether 'Yₙ converges in distribution to Y' implies 'Yₙ converges in probability to Y'. (15 marks)

हिंदी में प्रश्न पढ़ें

(a) माना X, Y, Z तीन परस्पर स्वतंत्र मानक घातीय चर हैं तथा W₁ = X + Y + Z, W₂ = (X + Y)/(X + Y + Z), W₃ = X/(X + Y). तब (i) W₁, W₂ एवं W₃ का संयुक्त बंटन निकालिए । (ii) W₁, W₂ एवं W₃ के सीमांत प्रायिकता घनत्व फलन ज्ञात कीजिए । (iii) W₁, W₂ एवं W₃ के परस्पर स्वतंत्र होने का परीक्षण कीजिए तथा इस पर अपनी टिप्पणी दीजिए । (10+6+4=20 अंक) (b) निम्नलिखित को सिद्ध या अस्वीकृत करने के लिए एक उदाहरण दीजिए : P(lim sup Aₙ) = 0 ⇒ Σₖ₌₁^∞ P(Aₖ) < ∞, जहाँ {Aₙ, n ≥ 1} घटनाओं की कोई श्रृंखला है जो कि संभाव्यता अंतराल (Ω, 𝓐, P) पर परिभाषित है । (15 अंक) (c) माना {Yₙ, n ≥ 1} यादृच्छिक चरों की एक श्रृंखला है तथा Y एक अपभ्रष्ट यादृच्छिक चर है । परीक्षण कीजिए कि क्या 'Yₙ बंटन में Y को अभिसरित होता है', से यह निष्कर्ष निकलता है कि 'Yₙ प्रायिकता में Y को अभिसरित होता है' । (15 अंक)

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Directive word: Derive

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How this answer will be evaluated

Approach

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).

Key points expected

  • 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

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness18%9Correctly identifies standard exponential PDFs, properly defines transformation regions with correct support constraints for (a); selects appropriate probability space and event construction for (b); correctly defines degenerate distribution and convergence modes for (c)Basic PDFs stated but support conditions vague; probability space mentioned but event construction unclear; convergence definitions present but impreciseWrong PDF forms, missing independence conditions, or confused definitions of convergence modes
Method choice22%11Uses Jacobian transformation method correctly for (a) with proper variable ordering; applies Borel-Cantelli lemma insight for (b); employs characteristic functions or direct CDF argument for (c) with degenerate limitAttempts transformation but Jacobian computation flawed; tries direct construction for (b) without Borel-Cantelli reference; uses definition but misses key simplification for degenerate caseWrong method entirely (e.g., MGF for (a) which fails here), or no clear method for any part
Computation accuracy24%12Accurate Jacobian determinant calculation, correct integration limits, proper PDF simplification to Gamma and Beta forms; precise probability calculations for counterexample; error-free limit computations for (c)Jacobian correct but arithmetic errors in simplification; counterexample numerically correct but verification incomplete; minor errors in limit argumentsMajor computational errors in Jacobian, wrong marginal distributions, or incorrect probability series evaluation
Interpretation20%10Clear explanation of independence structure showing W₁, W₂, W₃ mutually independent with statistical insight; explains why converse of Borel-Cantelli fails in (b); interprets equivalence of convergence modes for degenerate limits as special propertyStates independence result without insight; notes counterexample works without explaining Borel-Cantelli connection; mentions equivalence without explaining why degenerate case differsNo interpretation of results, or misinterprets dependence structure, or confuses convergence implications
Final answer & units16%8All PDFs explicitly stated with correct parameters and supports; counterexample fully specified with verification; proof concluded with clear statement of equivalence for degenerate casePDFs stated but parameters or support incomplete; counterexample given but verification sketchy; conclusion present but not sharply formulatedMissing final distributions, incomplete counterexample, or no conclusion for (c)

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