Statistics 2024 Paper II 50 marks Explain

Q3

(a) With respect to a given Linear Programming Problem (LPP), explain the following concepts : 15 marks (i) Extreme Point Solutions (ii) Duality Theorem (iii) Complementary Slackness Principle (b) Define a Transition Probability Matrix (TPM). When is it said to be Regular and Ergodic ? Check whether the following TPM is Regular or Ergodic. Hence or otherwise obtain the $\lim\limits_{n \to \infty} P^n$, where $P = \begin{pmatrix} 0.88 & 0.12 \\ 0.15 & 0.85 \end{pmatrix}$. 15 marks (c) The reliability function R(t) of a cutting assembly is given by : $$R(t) = \begin{cases} \left(1-\dfrac{t}{t_0}\right)^2, & 0 \leq t \leq t_0 \\ \quad 0 & , \quad t \geq t_0 \end{cases}$$ (i) Determine the failure rate. (ii) Does the failure rate increase or decrease with time ? (iii) Determine the mean time to failure. 8+4+8=20 marks

हिंदी में प्रश्न पढ़ें

(a) एक दी गई रैखिक प्रोग्रामन समस्या (एल पी पी) के संदर्भ में, निम्नलिखित संकल्पनाओं की व्याख्या कीजिए : 15 अंक (i) चरम बिन्दु समाधान (ii) द्वैत प्रमेय (iii) पूरक शिथिलता सिद्धांत (b) संक्रमण प्रायिकता मैट्रिक्स (टी पी एम) को परिभाषित कीजिए। इसे कब नियमित और अभ्यतिग्राय (एर्गोडिक) कहते हैं ? परीक्षण कीजिए कि क्या निम्नलिखित संक्रमण प्रायिकता मैट्रिक्स (टी पी एम) नियमित है या अभ्यतिग्राय (एर्गोडिक) है। इस प्रकार या अन्य प्रकार से $\lim\limits_{n \to \infty} P^n$ को प्राप्त कीजिए, जहाँ $P = \begin{pmatrix} 0.88 & 0.12 \\ 0.15 & 0.85 \end{pmatrix}$ है। 15 अंक (c) एक कतन समुच्चय (कटिंग असेंबली) का विश्वसनीयता फलन R(t) दिया गया है : $$R(t) = \begin{cases} \left(1-\dfrac{t}{t_0}\right)^2, & 0 \leq t \leq t_0 \\ \quad 0 & , \quad t \geq t_0 \end{cases}$$ (i) विफलता दर निर्धारित कीजिए। (ii) क्या विफलता दर समय के साथ बढ़ती या घटती है ? (iii) विफलता का औसत समय निर्धारित कीजिए। 8+4+8=20 अंक

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Directive word: Explain

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How this answer will be evaluated

Approach

Begin with clear definitions for (a)(i)-(iii) on LPP concepts, allocating ~30% time; for (b) define TPM, establish regularity/ergodicity criteria, then compute steady-state probabilities using eigenvalue or algebraic methods (~35% time); for (c) derive failure rate from R(t), analyze its monotonicity, and integrate for MTTF (~35% time). Structure: definitions → theorems → computational steps → physical interpretation.

Key points expected

  • (a)(i) Extreme Point Solutions: Definition as feasible region vertices, convex combination property, and Fundamental Theorem of LPP optimality at extreme points
  • (a)(ii) Duality Theorem: Statement of weak and strong duality, primal-dual relationship, and economic interpretation of shadow prices
  • (a)(iii) Complementary Slackness: Conditions relating primal slack and dual variables, optimality verification tool
  • (b) TPM definition with row-stochastic property; regularity (some P^n has all positive entries) vs ergodicity (irreducible + aperiodic); classification of given P as regular and ergodic; computation of limiting distribution π = (5/9, 4/9)
  • (c)(i) Failure rate λ(t) = 2/(t₀-t) for 0 ≤ t < t₀ using λ(t) = -R'(t)/R(t)
  • (c)(ii) Increasing failure rate (IFR) demonstration as λ'(t) > 0, indicating wear-out phase
  • (c)(iii) MTTF = t₀/3 via integration ∫₀^t₀ R(t)dt, with proper handling of improper integral at t₀

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Precise definitions for all three LPP concepts in (a); correct TPM properties stated; proper domain specification for R(t) in (c) with recognition of t₀ as parameterDefinitions present but imprecise (e.g., missing stochastic property of TPM); minor domain errors in R(t) setupMissing key definitional elements; confusion between primal and dual; incorrect TPM row/column sum properties
Method choice20%10Appropriate theorem selection: complementary slackness for optimality, eigenvalue/algebraic method for steady-state, standard reliability formulas for failure rate and MTTFCorrect methods chosen but inefficient (e.g., computing high powers of P instead of solving πP = π); minor formula errorsWrong method selection (e.g., simplex algorithm for steady-state); incorrect reliability formula application
Computation accuracy20%10Correct derivative for failure rate: 2/(t₀-t); accurate steady-state probabilities (5/9, 4/9); exact MTTF = t₀/3 with verified integrationMinor arithmetic errors in steady-state calculation; correct differentiation but algebraic slip in MTTF; correct trend but wrong coefficientMajor computational errors: wrong eigenvalues, incorrect failure rate formula, or integration mistakes yielding MTTF = t₀/2 or similar
Interpretation20%10Economic insight for duality (resource valuation); physical interpretation of regularity/ergodicity in Markov chains; clear IFR classification with engineering significance for cutting assembly wear-outSome interpretation present but superficial; mechanical classification without physical meaning; missing economic/engineering contextPurely computational with zero interpretation; no recognition of IFR implications for maintenance planning
Final answer & units20%10All six sub-parts answered with boxed/concluded results; limiting matrix explicitly stated; MTTF in time units matching t₀; clear statement of increasing failure rateAnswers present but poorly organized; missing explicit limiting matrix form; units implied but not statedIncomplete answers across parts; missing limiting distribution; no conclusion on failure rate trend; scattered results without sub-part labeling

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