Statistics 2024 Paper II 50 marks Compulsory Describe

Q1

(a) Consider a system consisting of three identical units connected in parallel. The unit reliability factor is 0·90. If the unit failures are independent of one another, and if the successful operation of the system depends on the satisfactory performance of any one unit, determine the system's reliability. 10 marks (b) Describe the procedure and some of the applications of Cumulative Sum (CUSUM) chart for monitoring process mean. 10 marks (c) Explain the following terms as used in sampling inspection plans : 5+5=10 marks (i) Producer's risk (ii) Average Outgoing Quality Limit (d) A Linear Programming Problem (LPP) in standard form is as given below : Optimize Z = CᵀX subject to AX = B with X ≥ 0 Write down the Dual Simplex form and its iterative procedure. 10 marks (e) What is Monte Carlo Simulation ? State the uses and applications of Monte Carlo Simulation. 10 marks

हिंदी में प्रश्न पढ़ें

(a) एक प्रणाली पर विचार कीजिए जिसमें तीन समान इकाइयाँ हों जो समानान्तर में जुड़ी हों। इकाई विश्वसनीयता कारक 0·90 है। यदि इकाई विफलताएँ एक दूसरे से स्वतंत्र हैं, और यदि प्रणाली का सफल संचालन किसी एक इकाई के संतोषजनक प्रदर्शन पर निर्भर करता है, तो प्रणाली की विश्वसनीयता ज्ञात कीजिए। 10 अंक (b) प्रक्रम माध्य की निगरानी के लिए संचयी योगफल (सी यू एस यू एम) चार्ट की कार्यविधि और उसके कुछ अनुप्रयोगों का वर्णन कीजिए। 10 अंक (c) प्रतिचयन निरीक्षण आयोजनाओं में उपयोग होने वाले निम्नलिखित पदों की व्याख्या कीजिए : 5+5=10 अंक (i) उत्पादक का जोखिम (ii) औसत निर्गमी गुणता सीमा (d) मानक रूप में एक रैखिक प्रोग्रामन समस्या (एल पी पी) नीचे दी गई है : इष्टतमीकरण (ऑप्टिमाइज़) कीजिए Z = CᵀX निम्न प्रतिबन्ध के अन्तर्गत AX = B साथ में X ≥ 0 द्वैत एकल रूप और इसकी पुनरावृत्त प्रक्रिया को लिखिए। 10 अंक (e) मोंटे कार्लो अनुकरण क्या है ? मोंटे कार्लो अनुकरण के उपयोगों तथा अनुप्रयोगों को बताइए। 10 अंक

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Directive word: Describe

This question asks you to describe. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.

See our UPSC directive words guide for a full breakdown of how to respond to each command word.

How this answer will be evaluated

Approach

The directive 'describe' demands systematic exposition of procedures and concepts across all five sub-parts. Allocate approximately 15-18 minutes to part (a) requiring calculation, 12-15 minutes each to descriptive parts (b), (c), and (e), and 10-12 minutes to part (d) on dual simplex. Structure as: direct calculation for (a); stepwise procedural description for (b) with industrial applications; precise definitions with formulas for (c); algorithmic presentation for (d); and conceptual definition followed by domain-specific applications for (e).

Key points expected

  • Part (a): Correct application of parallel system reliability formula R_system = 1 - (1 - 0.90)³ = 1 - (0.10)³ = 0.999 or 99.9%
  • Part (b): CUSUM chart construction using V-mask or decision interval, cumulative sum calculation S_i = Σ(x_j - μ₀), and applications in pharmaceutical quality control or ISRO component manufacturing
  • Part (c)(i): Producer's risk (α) as probability of rejecting good lot (AQL quality), typically set at 5% with Type I error interpretation
  • Part (c)(ii): AOQL as maximum average outgoing quality after rectifying inspection, formula AOQ = p·P_a·(N-n)/N for sampling with replacement
  • Part (d): Dual simplex form with primal maximization converting to dual minimization, conditions for optimality (all c_j - z_j ≤ 0), and iterative steps for pivot selection when b_i < 0
  • Part (e): Monte Carlo as stochastic simulation using random number generation, with applications in nuclear shielding design, financial risk modeling, or Indian monsoon prediction

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly identifies parallel system structure for (a), specifies target mean μ₀ and reference value K for CUSUM in (b), defines AQL and LTPD contexts for (c), establishes primal-dual correspondence with correct inequality directions for (d), and identifies random sampling from probability distributions as core Monte Carlo mechanism for (e)Partially correct setups with minor errors in system identification or missing reference parameters; vague dual simplex conditionsConfuses parallel with series reliability, omits key parameters like V-mask parameters, or fundamentally misunderstands primal-dual relationship
Method choice20%10Selects complementary probability method for parallel reliability, uses decision interval or V-mask CUSUM algorithm, applies correct AOQ formulas with rectifying inspection, employs standard primal-dual transformation rules, and specifies inverse transform or acceptance-rejection sampling for Monte CarloCorrect but inefficient methods; acceptable CUSUM description without algorithmic detail; incomplete dual simplex pivot rulesIncorrect method selection such as direct multiplication for parallel systems, confused OC curve concepts, or wrong simplex variant identification
Computation accuracy20%10Precise calculation: 1 - 0.001 = 0.999 for (a); correct cumulative sum recursion; accurate AOQ formula application with (N-n)/N factor; proper dual simplex tableau operations; appropriate sample size considerations for Monte Carlo precisionMinor arithmetic slips like 0.999 vs 0.99; acceptable formula presentation with calculation errors; incomplete tableau demonstrationGross computational errors, incorrect probability calculations, or missing essential computational steps across multiple parts
Interpretation20%10Interprets 0.999 reliability as substantial improvement over single unit; explains CUSUM's sensitivity to small shifts vs Shewhart charts; contrasts producer's and consumer's risks; explains dual simplex feasibility restoration; evaluates Monte Carlo convergence and variance reduction techniquesBasic interpretation without comparative analysis; standard definitions without insight into relative advantages; limited convergence discussionNo interpretation of numerical results; fails to distinguish method advantages; omits practical significance of calculated values
Final answer & units20%10Clear final answers: R_system = 0.999 or 99.9% for (a); complete CUSUM procedure summary; α = 0.05 with AOQL formula; dual simplex optimality conditions; Monte Carlo definition with 3-4 specific Indian applications (census estimation, railway scheduling, agricultural forecasting, nuclear physics)Correct answers with inconsistent formatting; partial application lists; missing units or probability expressionsMissing final answers, incorrect units, or incomplete response to any sub-part exceeding 2 marks loss

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