Q1
(a) Two events A and B are such that P(A) = 1/3, P(B) = 1/4 and P(A|B) + P(B|A) = 2/3. Evaluate the following: (i) P(A^c ∪ B^c) (5 marks) (ii) P(A|B^c) + P(B|A^c) (5 marks) (b) Suppose the joint probability function of two random variables X and Y is f(x, y) = (xy^(x-1))/3; x = 1, 2, 3 and 0 < y < 1. Compute the following: (i) P(X ≥ 2 and Y ≥ 1/2) (5 marks) (ii) P(X ≥ 2) (5 marks) (c) Let X₁, X₂, ... is a sequence of independent and identically distributed random variables with mean (μ) and variance (σ²) < ∞, and assume Sₙ = X₁ + X₂ + ... + Xₙ. Show that WLLN does not hold for sequence ⟨Sₙ⟩ of random variables. (10 marks) (d) Write the criterion of a good estimator. Let X₁, X₂ be iid P(λ) random variables, then show that T = X₁ + X₂ is sufficient while T = X₁ + 2X₂ is not sufficient for estimating λ. (10 marks) (e) For testing H₀: μ = 100 vs. H₁: μ ≠ 100, a random sample of size 50 is drawn from a normal population with unknown mean μ and variance 200. If α = 0.05, then obtain the critical region. (10 marks)
हिंदी में प्रश्न पढ़ें
(a) दो घटनाएँ A और B इस प्रकार हैं कि P(A) = 1/3, P(B) = 1/4 और P(A|B) + P(B|A) = 2/3. निम्नलिखित के मान निकालिए : (i) P(A^c ∪ B^c) (5) (ii) P(A|B^c) + P(B|A^c) (5) (b) मान लीजिए कि दो यादृच्छिक चरों X और Y का संयुक्त प्रायिकता फलन f(x, y) = (xy^(x-1))/3; x = 1, 2, 3 और 0 < y < 1 है। निम्नलिखित का परिकलन कीजिए : (i) P(X ≥ 2 और Y ≥ 1/2) (5) (ii) P(X ≥ 2) (5) (c) मान लीजिए कि X₁, X₂, ... स्वतंत्र और सर्वसम बाँटित यादृच्छिक चरों का एक अनुक्रम है, जिसका माध्य (μ) और प्रसरण (σ²) < ∞ है, तथा मान लीजिए कि Sₙ = X₁ + X₂ + ... + Xₙ है। दर्शाइए कि यादृच्छिक चरों का अनुक्रम ⟨Sₙ⟩ दुर्बल बहुत संख्या नियम (WLLN) का पालन नहीं करता है। (10) (d) एक अच्छे आकलक का मापदंड लिखिए। माना कि X₁, X₂ स्वतंत्र और सर्वसम बाँटित (iid) P(λ) यादृच्छिक चर हैं, तब दर्शाइए कि λ के आकलन के लिए T = X₁ + X₂ पर्याप्त है, जबकि T = X₁ + 2X₂ पर्याप्त नहीं है। (10) (e) H₀: μ = 100 विरुद्ध H₁: μ ≠ 100 के परीक्षण के लिए अज्ञात माध्य μ और प्रसरण 200 वाली एक प्रसामान्य समष्टि से आमाप 50 का एक यादृच्छिक प्रतिदर्श लिया गया है। यदि α = 0.05 है, तो कांतिक क्षेत्र प्राप्त कीजिए। (10)
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Approach
Solve this multi-part numerical problem by allocating time proportionally to marks: approximately 20% for (a), 20% for (b), 20% for (c), 20% for (d), and 20% for (e). Begin each sub-part by stating the relevant formula or theorem, show complete step-by-step working, and conclude with clearly boxed final answers. For (c) and (d), include brief theoretical justification before numerical demonstration.
Key points expected
- For (a): Use P(A|B) + P(B|A) = 2/3 to find P(A∩B) = 1/12, then apply De Morgan's laws for P(A^c ∪ B^c) = 1 - P(A∩B) = 11/12
- For (a)(ii): Calculate conditional probabilities using P(A|B^c) = [P(A) - P(A∩B)]/P(B^c) and similar for P(B|A^c), yielding 1/9 + 1/8 = 17/72
- For (b): Integrate joint PDF over appropriate regions; for (i) sum x=2,3 and integrate y from 1/2 to 1; for (ii) marginalize over y
- For (c): Show WLLN fails by proving Var(S_n)/n² → σ² ≠ 0, so S_n/n does not converge in probability to μ (unlike sample mean)
- For (d): State sufficiency criterion (Factorization theorem), show T=X₁+X₂ ∼ P(2λ) allows factorization, while T=X₁+2X₂ does not
- For (e): Construct z-test with σ²=200, n=50; critical region |z| > 1.96 becomes |x̄ - 100| > 3.92 or x̄ < 96.08 or x̄ > 103.92
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies all given parameters, states appropriate theorems (De Morgan's, Factorization, CLT/WLLN, Neyman-Pearson), and sets up proper probability spaces for each sub-part | Identifies most parameters correctly but misses some setup details or uses slightly inappropriate theorem statements | Major errors in identifying given information, wrong theorems stated, or fundamentally incorrect probability space setup |
| Method choice | 20% | 10 | Selects optimal methods: intersection formula for (a), integration/summation for (b), variance analysis for (c), Factorization theorem for (d), z-test construction for (e); all methods are efficient and mathematically sound | Uses acceptable but suboptimal methods, or correct methods with minor inefficiencies; may use definition instead of theorem for sufficiency | Wrong methods chosen (e.g., assumes independence in (a), uses discrete sum instead of integral in (b), confuses WLLN with SLLN in (c)) |
| Computation accuracy | 20% | 10 | All calculations precise: P(A∩B)=1/12, (a)(i)=11/12, (a)(ii)=17/72, (b) integrals evaluate correctly, (e) critical values exactly 96.08 and 103.92; no arithmetic errors | Minor arithmetic slips (e.g., fraction addition errors, integration constants mishandled) but methodologically correct approach | Major computational errors, wrong final values, or algebraic mistakes that propagate through solutions |
| Interpretation | 20% | 10 | Explains why WLLN fails for ⟨S_n⟩ vs holds for ⟨S_n/n⟩; interprets sufficiency intuitively (T captures all λ-information); explains critical region's meaning for hypothesis testing | Brief interpretation provided but lacks depth or misses connection between theoretical result and practical implication | No interpretation provided, or misinterprets results (e.g., claims WLLN holds for S_n, confuses sufficient with unbiased) |
| Final answer & units | 20% | 10 | All six sub-part answers clearly stated and boxed: (a)(i) 11/12, (a)(ii) 17/72, (b) numerical values, (c) proved statement, (d) proved statements, (e) critical region explicitly defined; proper probability notation throughout | Most answers present but some unclear or unboxed; minor notation inconsistencies | Missing answers, wrong values presented confidently, or answers without proper probability/measurement units where applicable |
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