Statistics 2022 Paper I 50 marks Solve

Q3

(a) (i) How large a sample must be taken in order that the probability will be at least 0·90 that the sample mean will be within 0·4 – neighbourhood of the population mean, provided the population standard deviation is 2 ? (8 marks) (ii) Examine whether the weak law of large numbers holds for the sequence {Xₖ} of independent random variables defined as follows : $$P(X_k = -1 - \frac{1}{k}) = \frac{1}{2}\left\{1 - \left(1 - \frac{1}{k^2}\right)^{1/2}\right\},$$ $$P(X_k = 1 + \frac{1}{k}) = \frac{1}{2}\left\{1 + \left(1 - \frac{1}{k^2}\right)^{1/2}\right\}.$$ (7 marks) (b) Theoretical probabilities in the four cells of a multinomial distribution are $\frac{2+\theta}{4}$, $\frac{1-\theta}{4}$, $\frac{1-\theta}{4}$ and $\frac{\theta}{4}$, whereas the observed frequencies are 108, 27, 30 and 8 respectively, then estimate θ by maximum likelihood method. Also, obtain the standard error of the estimate. (20 marks) (c) If X is a random variable with characteristic function $$\varphi(t) = \begin{cases} 1-|t|, & |t| \leq 1 \\ 0, & \text{otherwise}, \end{cases}$$ then obtain the corresponding probability density function. (15 marks)

हिंदी में प्रश्न पढ़ें

(a) (i) एक प्रतिदर्श कितना बड़ा लेना चाहिए ताकि इस बात की प्रायिकता कम-से-कम 0·90 होगी कि प्रतिदर्श माध्य समष्टि माध्य के 0·4 - सामीप्य के दायरे में होगा, बशर्ते कि समष्टि मानक विचलन 2 है ? (8 अंक) (ii) परीक्षण कीजिए कि क्या बहुत संख्याओं का दुर्बल नियम निम्न परिभाषित स्वतंत्र यादृच्छिक चरों के अनुक्रम {Xₖ} के लिए लागू होता है : $$P(X_k = -1 - \frac{1}{k}) = \frac{1}{2}\left\{1 - \left(1 - \frac{1}{k^2}\right)^{1/2}\right\},$$ $$P(X_k = 1 + \frac{1}{k}) = \frac{1}{2}\left\{1 + \left(1 - \frac{1}{k^2}\right)^{1/2}\right\}.$$ (7 अंक) (b) एक बहुपद बंटन में चार कोष्ठकों की सैद्धांतिक प्रायिकताएं $\frac{2+\theta}{4}$, $\frac{1-\theta}{4}$, $\frac{1-\theta}{4}$ और $\frac{\theta}{4}$ हैं, जबकि प्रेक्षित बारंबारताएं क्रमशः: 108, 27, 30 और 8 हैं, तब θ का आकलन अधिकतम सम्भाविता विधि से कीजिए। आकल की मानक त्रुटि भी निकालिए। (20 अंक) (c) यदि X एक यादृच्छिक चर है जिसका अभिलक्षण फलन $$\varphi(t) = \begin{cases} 1-|t|, & |t| \leq 1 \\ 0, & \text{अन्यथा}, \end{cases}$$ है, तब संगत प्रायिकता घनत्व फलन को प्राप्त कीजिए। (15 अंक)

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How this answer will be evaluated

Approach

Solve this multi-part numerical problem by allocating approximately 15 minutes to part (a)(i) on sample size determination using CLT, 15 minutes to part (a)(ii) on verifying WLLN conditions, 25 minutes to part (b) on MLE estimation and standard error computation for multinomial data, and 20 minutes to part (c) on deriving PDF from characteristic function via Fourier inversion. Begin each part with clear statement of the statistical principle being applied, show all computational steps explicitly, and conclude with precise numerical answers or definitive conclusions.

Key points expected

  • Part (a)(i): Apply Central Limit Theorem with z₀.₉₀ = 1.645 to obtain n ≥ (1.645 × 2/0.4)² = 67.65 → n = 68
  • Part (a)(ii): Verify E(Xₖ) = 1/k and Var(Xₖ) = 1 - 1/k², then apply Chebyshev or Kolmogorov's criterion to establish WLLN holds
  • Part (b): Formulate multinomial likelihood L(θ), take log-likelihood, solve dℓ/dθ = 0 to get θ̂ = (2×108 - 27 - 30 + 2×8)/(108+27+30+8) = 0.5, then compute Fisher information for SE(θ̂)
  • Part (c): Apply Fourier inversion formula f(x) = (1/2π)∫₋₁¹ (1-|t|)e⁻ⁱᵗˣ dt, evaluate to obtain f(x) = (1/πx²)(1 - cos x) = (1/2π)sinc²(x/2) for x ≠ 0, with f(0) = 1/2π
  • Demonstrate understanding that characteristic function φ(t) = (1-|t|)₊ corresponds to triangular distribution on [-1,1] in frequency domain, yielding Fejér kernel/sinc² in density domain

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly identifies CLT applicability for (a)(i), recognizes need to check E(Xₖ)→0 and ΣVar(Xₖ)/k² < ∞ for WLLN in (a)(ii), properly sets up multinomial likelihood with cell probabilities summing to 1 for (b), and correctly states Fourier inversion theorem with proper integration limits for (c)Identifies correct broad approach for each part but makes minor errors in probability setup (e.g., wrong z-value, incomplete WLLN conditions, or incorrect likelihood formulation)Misidentifies the statistical framework (e.g., uses Chebyshev directly without CLT for sample size, confuses WLLN with SLLN conditions, or attempts method of moments instead of MLE)
Method choice20%10Selects optimal solution paths: exact normal sample size formula for (a)(i), Kolmogorov's criterion or direct variance verification for WLLN in (a)(ii), standard multinomial MLE derivation with Fisher information for (b), and elegant contour/trigonometric evaluation of Fourier integral for (c)Uses acceptable but suboptimal methods (e.g., Chebyshev inequality giving loose bound for sample size, or numerical integration approach for characteristic function inversion)Chooses inappropriate methods (e.g., Poisson approximation for sample size, assumes identical distribution for WLLN verification, or attempts differentiation under integral sign without justification)
Computation accuracy20%10Executes all calculations flawlessly: n = 68 (not 67), correct algebraic simplification of E(Xₖ) and Var(Xₖ) with proper limit behavior, exact MLE θ̂ = 0.5 with correct Fisher information I(θ) = n/(2+θ)(1-θ) + n/θ(1-θ) evaluated at θ̂, and precise evaluation of ∫(1-|t|)cos(tx)dt = 2(1-cos x)/x²Minor computational slips (e.g., rounding n to 67, arithmetic errors in variance calculation, or sign errors in Fisher information) that don't fundamentally invalidate the approachMajor computational errors (e.g., incorrect z-value, failure to simplify complex expressions, wrong final answer for MLE, or inability to evaluate the Fourier integral)
Interpretation20%10Provides insightful interpretation: explains why n=68 guarantees 90% confidence for (a)(i), discusses rate of convergence to zero mean for WLLN in (a)(ii), interprets MLE θ̂=0.5 as indicating equal preference patterns in multinomial data for (b), and recognizes the derived PDF as the Fejér kernel appearing in spectral analysis and Cesàro summation for (c)States correct conclusions without deeper insight, or provides generic interpretations not specific to the numerical results obtainedMisinterprets results (e.g., claims WLLN fails when it holds, draws incorrect inference about population from MLE, or fails to recognize the characteristic function-PDF relationship)
Final answer & units20%10Presents all final answers with precision: n = 68 (sample size, dimensionless), explicit WLLN verification with concluding statement, θ̂ = 0.5 ± SE with numerical standard error value, and closed-form PDF f(x) = (1-cos x)/(πx²) for x≠0, 1/2π for x=0; all boxed or clearly demarcatedCorrect final answers but poorly formatted, missing units where relevant, or incomplete presentation of standard errorMissing final answers, incorrect numerical values, or failure to specify the PDF at x=0 as a separate case

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