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Statistics 2024 Paper II 50 marks Explain

Paper II — Q3

(a) With respect to a given Linear Programming Problem (LPP), explain the following concepts : 15 marks (i) Extreme Point…

(a)
(i)

With respect to a given Linear Programming Problem (LPP), explain the following concepts : 15 marks 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 limlimitsₙ → ∞ Pⁿ, where P = 0.88 & 0.12 0.15 & 0.85 . 15 marks

(c)
(i)

The reliability function R(t) of a cutting assembly is given by : R(t) = (1-t/(t₀))², & 0 ≤ t ≤ t₀ 0 & , t ≥ t₀ 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)
(i)

एक दी गई रैखिक प्रोग्रामन समस्या (एल पी पी) के संदर्भ में, निम्नलिखित संकल्पनाओं की व्याख्या कीजिए : 15 अंक 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 limlimitsₙ → ∞ Pⁿ को प्राप्त कीजिए, जहाँ P = 0.88 & 0.12 0.15 & 0.85 है। 15 अंक

(c)
(i)

एक कतन समुच्चय (कटिंग असेंबली) का विश्वसनीयता फलन R(t) दिया गया है : R(t) = (1-t/(t₀))², & 0 ≤ t ≤ t₀ 0 & , t ≥ t₀ 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

Q3 of the 2024 UPSC Mains Statistics Paper II, as printed
The question as printed in the 2024 Statistics paper
Evaluate your answer to this question → See all 2024 Statistics questions

Model answer

Written by UPSC Answer Check against this question's marking rubric, to the expected length. UPSC does not publish answers for Mains — this is one way to score well, not an official key.

(a) Concepts in Linear Programming

(i) Extreme Point Solutions In an LPP, the feasible region defined by the constraints is a convex polyhedron. A feasible solution is an extreme point (vertex) if it cannot be expressed as a strict convex combination of two other distinct feasible solutions, i.e. it lies at a corner where at least n linearly independent constraints intersect (n being the number of variables). The Fundamental Theorem of LPP states that if an optimal solution exists, at least one optimal solution occurs at an extreme point, because a linear objective function attains its maximum/minimum over a convex polytope at a vertex. This is why the simplex method searches only among basic feasible solutions (extreme points) rather than the infinite feasible set.

(ii) Duality Theorem Every primal LPP has an associated dual. Weak duality states that for any feasible primal x and dual y, the primal objective ≤ dual objective (for maximisation primal). Strong duality states that if either problem has a finite optimal solution, so does the other, and their optimal objective values are equal. The dual variables are shadow prices: they measure the marginal change in the optimal objective per unit relaxation of a constraint, giving the economic worth of scarce resources.

(iii) Complementary Slackness Principle At optimality, for each primal constraint, the product of its slack and the corresponding dual variable is zero; similarly for each dual constraint, the product of its surplus and the corresponding primal variable is zero. Thus a resource with positive slack (not fully used) has zero shadow price, and a resource with positive shadow price is fully exhausted. It is a powerful optimality-verification tool.

(b) Transition Probability Matrix A TPM P = [p_ij] gives the one-step transition probabilities of a Markov chain, satisfying p_ij ≥ 0 and each row summing to 1 (row-stochastic). A chain is Regular if some power Pⁿ has all entries strictly positive. It is Ergodic if it is irreducible (all states communicate) and aperiodic; a regular chain is ergodic.

For P = [[0.88,0.12],[0.15,0.85]], all entries are positive, so P itself has all positive entries; hence P is regular and ergodic. The limiting distribution π satisfies πP = π, π₁+π₂=1: 0.88π₁+0.15π₂ = π₁ → 0.15π₂ = 0.12π₁ → π₁/π₂ = 5/4. With π₁+π₂=1: π = (5/9, 4/9). Hence lim Pⁿ has both rows equal to (5/9, 4/9).

(c) Reliability Function R(t) = (1 − t/t₀)², 0 ≤ t ≤ t₀; 0 for t ≥ t₀.

(i) Failure rate: λ(t) = −R′(t)/R(t). R′(t) = −(2/t₀)(1 − t/t₀). So λ(t) = [(2/t₀)(1 − t/t₀)] / (1 − t/t₀)² = 2/(t₀ − t), for 0 ≤ t < t₀.

(ii) Monotonicity: λ′(t) = 2/(t₀ − t)² > 0, so the failure rate increases with time — an Increasing Failure Rate (IFR) distribution, characteristic of a wear-out phase.

(iii) MTTF: MTTF = ∫₀^t₀ R(t)dt = ∫₀^t₀(1 − t/t₀)²dt. Let u = 1 − t/t₀, dt = −t₀du: = t₀∫₀¹u²du = t₀/3.

Thus the assembly has an increasing failure rate and a mean life of one-third of its maximum support t₀.

What "Explain" is asking you to do

Make the working of something clear — what sets it off, what follows from what, and what it produces. Explain is the Commission's mechanism word: it dominates the technical papers and the “explain why” stems, where the marks sit in the causal chain and not in the label.

Structure that answers it

State what it is → the initiating condition → the chain of cause, step by step → an instance where it plays out → what the chain produces

Where marks are lost

Describing what something looks like instead of why it works that way. Naming the stages without linking them reads as description too.

All UPSC directive words, compared →

How this answer will be evaluated

Approach

(a) explain: definition/context > points in order > small example > short close | (b) define: precise definition > the distinguishing feature > one example | (c(i)) calculate: given > formula > substitution > result with units > interpretation | (c(ii)) comment: context > arguments both sides > judgment > close | (c(iii)) calculate: given > formula > substitution > result with units > interpretation Full marks: All parts fully answered with correct derivations, clear definitions, and proper interpretation.

Key points expected

  • Define Extreme Point Solutions as vertices of feasible region
  • State Duality Theorem linking primal and dual optimal values
  • State Complementary Slackness Principle conditions for optimality
  • Provide a small example or context for each concept
  • Define TPM with row-stochastic property
  • Define Regular (some power has all positive entries)
  • Define Ergodic (unique stationary distribution)
  • Calculate limit P^n as n approaches infinity

Evaluation rubric

Each sub-part is marked on its own, against the marks and word limit printed on the paper.

  1. (a) Define and explain three specific LPP concepts. 15 marks

    explain— definition/context → points in order → small example → short close

    Must cover

    • Define Extreme Point Solutions as vertices of feasible region
    • State Duality Theorem linking primal and dual optimal values
    • State Complementary Slackness Principle conditions for optimality
    • Provide a small example or context for each concept

    Loses marks

    • Confusing extreme points with basic feasible solutions
    • Stating duality without mentioning optimality conditions
    • Omitting the 'slackness' condition in complementary slackness

    Earns more

    • Mention geometric interpretation of extreme points
    • Note strong duality theorem specifically
    • Show algebraic form of complementary slackness

    Extra mark

    • Reference to Simplex method for finding extreme points
    • Mention sensitivity analysis in context of duality
  2. (b) Define TPM, Regular/Ergodic, classify given matrix, find limit. 15 marks

    define— precise definition → the distinguishing feature → one example

    Must cover

    • Define TPM with row-stochastic property
    • Define Regular (some power has all positive entries)
    • Define Ergodic (unique stationary distribution)
    • Calculate limit P^n as n approaches infinity

    Loses marks

    • Confusing Regular with Irreducible
    • Failing to check if the given matrix is regular
    • Incorrect calculation of the limit matrix

    Earns more

    • Show calculation of stationary distribution vector
    • Verify regularity by checking P^2 or P^k
    • State that regular implies ergodic for finite chains

    Extra mark

    • Mention Perron-Frobenius theorem
    • Provide a diagram of the state transition graph
  3. (c(i)) Determine the failure rate function from R(t). 8 marks

    calculate— given → formula → substitution → result with units → interpretation

    Must cover

    • State formula for failure rate h(t) = -R'(t)/R(t)
    • Differentiate R(t) = (1 - t/t0)^2
    • Substitute into formula to find h(t)
    • Simplify to h(t) = 2/(t0 - t)

    Loses marks

    • Using h(t) = f(t)/R(t) without deriving f(t)
    • Sign error in derivative of R(t)
    • Failing to simplify the final expression

    Earns more

    • Show step-by-step differentiation
    • State domain of validity 0 <= t < t0

    Extra mark

    • Mention units of failure rate (per time unit)
  4. (c(ii)) State if failure rate increases or decreases with time. 4 marks

    comment— context → arguments both sides → judgment → close

    Must cover

    • Analyze h(t) = 2/(t0 - t) for t in [0, t0)
    • State that h(t) increases as t increases
    • Justify by noting denominator decreases as t increases

    Loses marks

    • Stating 'decreases' without justification
    • Confusing failure rate with reliability function

    Earns more

    • Mention this is an increasing failure rate (IFR) distribution
    • Relate to wear-out phase of bathtub curve

    Extra mark

    • Sketch the shape of the h(t) curve
  5. (c(iii)) Determine the mean time to failure (MTTF). 8 marks

    calculate— given → formula → substitution → result with units → interpretation

    Must cover

    • State formula MTTF = integral of R(t) from 0 to infinity
    • Set up integral of (1 - t/t0)^2 from 0 to t0
    • Perform integration to get t0/3
    • State final answer with units

    Loses marks

    • Integrating from 0 to infinity without splitting at t0
    • Arithmetic error in integration
    • Forgetting to square the term before integrating

    Earns more

    • Show substitution u = 1 - t/t0 for integration
    • Verify result using alternative method if possible

    Extra mark

    • Mention that MTTF is the first moment of failure time

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