UPSC Statistics 2022 — Paper II

Distinguish requires clear differentiation followed by explanation. Allocate approximately 20% time to part (a) on process vs product control with variation sources, 25% to part (b) simulation with proper random number mapping, 15% to part (c) conditional probability derivation for Poisson, 20% to part (d) game theory assumptions and algebraic method, and 20% to part (e) censoring importance and MLE for exponential Type-2. Begin with definitions, proceed to analytical derivations or computational steps, and conclude with interpretations.

• Part (a): Clear distinction between process control (monitoring during production) and product control (acceptance sampling); identification of chance causes (random, inherent) and assignable causes (special, identifiable) of variation; remedial measures for each source
• Part (b): Correct probability interval mapping for selling price (00-19, 20-69, 70-99), variable cost (00-29, 30-89, 90-99), and sales volume (00-29, 30-59, 60-99); proper profit calculation as (Price - Variable Cost) × Volume - Fixed Cost; accurate simulation table with 10 trials and average profit computation
• Part (c): Derivation of P(N(s)=k|N(t)=n) using independent increments property resulting in Binomial(n, s/t) distribution; recognition that conditional distribution depends only on ratio s/t not on rate parameter λ
• Part (d): Assumptions: two players, finite strategies, zero-sum, simultaneous moves, complete information; maximin (player A's security level) and minimax (player B's security level) principles; algebraic method using mixed strategies with probability variables p and q, solving simultaneous equations for value of game
• Part (e): Importance of censoring: time/cost efficiency, ethical considerations, handling heavy-tailed distributions; Type-2 censoring with r failures out of n items; MLE derivation for θ with estimator T/r where T is total time on test, showing unbiasedness and variance properties

This is a numerical problem requiring systematic calculation across three parts. Allocate approximately 35% time to part (a) with its four sub-parts on control charts (15 marks), 30% to part (b) on Weibull distribution derivations (15 marks), and 35% to part (c) on acceptance sampling analysis (20 marks). Begin each part with clear identification of given parameters, show all formulas before substitution, and conclude with explicit interpretation of results.

• Part (a)(i): Correct application of x̄ chart limits using A₂R̄ and R chart limits using D₃R̄, D₄R̄ with n=5
• Part (a)(ii): Estimation of process standard deviation using σ̂ = R̄/d₂ = 4.56/2.326
• Part (a)(iii): Calculation of Cp and Cpk indices comparing process capability with specification limits 19±5
• Part (a)(iv): Calculation of β-risk (Type II error) using normal distribution for shifted mean μ=24
• Part (b): Derivation of Weibull hazard function h(t) = (β/α)(t/α)^(β-1) and reliability R(t) = exp[-(t/α)^β], with IFR/CFR/DFR classification based on β
• Part (c): Analysis of variable sample size plan, calculation of acceptance probability using binomial/Poisson approximation, and evaluation against LTPD=0.05 for consumer protection

This is a multi-part numerical problem requiring you to solve three distinct operations research scenarios. Allocate approximately 40% of time to part (a) given its 20 marks, and 30% each to parts (b) and (c). Begin with clear problem identification for each sub-part, show all working steps with proper formulae, and conclude with precise numerical answers with units. For part (b), balance theoretical explanation with mathematical derivation.

• Part (a): Calculate annual wastage rates from cumulative percentages, determine survival probabilities, compute required annual recruitment using renewal equation, and find average service length for promotion using weighted probability distribution
• Part (a): Correctly interpret 'total percentage who have left' as cumulative distribution and derive conditional probabilities of leaving in each specific year
• Part (b): Define queuing system components (arrival process, service mechanism, queue discipline) and derive steady-state probabilities for M/M/1 using balance equations with ρ = λ/μ < 1
• Part (b): Obtain explicit formulas for P₀ = 1-ρ, Pₙ = ρⁿ(1-ρ), and derive busy period distribution using Takács formula or generating function approach
• Part (c): Apply EOQ model with D = 400,000 metres/year, C₀ = ₹2,700, Cₕ = ₹60/metre/year, calculate optimal Q*, cycle time, and total minimum cost
• Part (c): Distinguish cost minimization from profit maximization by incorporating revenue function, show that optimal quantity remains unchanged under constant price, and compute gross profit as (360-240) × 400,000

Begin with the directive 'solve' for part (a), applying the Big-M penalty method to convert equality constraints and maximize the objective. Allocate approximately 40% of time to part (a) given its 20 marks, 30% to part (b) for crew scheduling optimization, and 30% to part (c) for sequential sampling theory and design. Structure as: (a) complete LP solution with simplex iterations, (b) layover time matrix and optimal pairing, (c) definition followed by ASN and OC curve construction.

• Part (a): Convert to standard form using Big-M penalty for equality constraints; introduce artificial variables A₁, A₂ with -M coefficient in objective
• Part (a): Execute simplex iterations showing entering and leaving variables until optimality reached with Z_max = 15, x₁=2.5, x₂=5, x₃=2.5, x₄=0
• Part (b): Construct 4×4 layover time matrix for Delhi-based and Jaipur-based crews; calculate layovers respecting 5-hour minimum
• Part (b): Identify optimal pairings minimizing total layover: Flight 1-101 (Delhi base), 2-102 (Delhi), 3-103 (Jaipur), 4-104 (Jaipur)
• Part (c): Define sequential sampling as item-by-item inspection with decision boundaries; state Wald's SPRT principles
• Part (c): Calculate decision parameters h₁, h₂, s and construct acceptance/rejection lines; provide ASN ~ 40-50 and OC curve characteristics

This multi-part question requires precise calculation and derivation across five distinct statistical domains. Allocate approximately 20% time to each sub-part: (a) construct link relatives table and seasonal indices using chain relatives method; (b) derive OLS estimators' properties using Gauss-Markov assumptions; (c) set up 2SLS normal equations showing projection onto instruments; (d) apply linear survivorship assumption to solve for l(41); (e) compute percentile ranks then transform to T-scores using given normal table. Present each part clearly with step-by-step working.

• For (a): Calculate chain relatives by expressing each quarter's value as percentage of preceding quarter, then obtain corrected relatives and seasonal indices normalized to 400
• For (b): Derive E(α̂) = α and E(β̂) = β showing unbiasedness, then derive Var(α̂) = σ²ΣX²/(nΣx²) and Var(β̂) = σ²/Σx² using matrix or scalar algebra
• For (c): State first stage projection Ŷ₁ = X(X'X)⁻¹X'Y₁, then second stage OLS of y on Ŷ₁ and X₁ to obtain 2SLS estimator β̂₂ₛₗₛ = (Z'PₓZ)⁻¹Z'Pₓy where Z = [Y₁|X₁]
• For (d): Use linearity of l(x) to establish e°₄₀ = ½ + l(41)/l(40) × e°₄₁, then solve l(41) = l(40)(e°₄₀ - ½)/(e°₄₁ + ½) and compute numerical value
• For (e): Compute cumulative frequencies, percentile ranks P = (100/N)(C - ½), find corresponding z-scores from normal table, then T = 50 + 10z for each score value

Explain the theoretical foundations across all three parts with appropriate mathematical derivations where required. Allocate approximately 30% effort to part (a) on AIC and ARMA order selection, 40% to part (b) on autocorrelation and MLE estimation given its higher marks, and 30% to part (c) on industrial data collection methods. Structure with clear sectional headings, begin each part with precise definitions, develop through step-by-step reasoning, and conclude with practical implications or limitations.

• Part (a): Definition of AIC as -2log(L) + 2k where L is likelihood and k is number of parameters; trade-off between goodness-of-fit and model complexity; comparison with BIC/AICc; application to ARMA(p,q) via minimization over candidate orders
• Part (b): Autocorrelation coefficient ρ_k = Cov(u_t, u_{t-k})/Var(u_t); consequences for OLS: biased standard errors, inefficient estimates, invalid t/F tests; MLE derivation for AR(1) errors with transformation matrix Ω, concentrated likelihood, and iterative estimation
• Part (c): Methods: census vs sample surveys, ASI (Annual Survey of Industries) schedule, establishment surveys; Official publications: ASI Summary Results, Index of Industrial Production (IIP), Economic Census; Agencies: CSO (now NSO), DIPP, Labour Bureau, RBI industrial data
• Mathematical rigor: Proper likelihood functions, matrix notation for GLS transformation, stationarity conditions for AR(1) parameter |ρ| < 1
• Applied context: Indian industrial statistics system, ASI coverage of registered manufacturing, limitations of informal sector data

The directive 'explain' demands conceptual clarity with logical exposition. For part (a) carrying 20 marks, allocate ~40% effort covering CDR, ASDR, IMR, MMR with direct and indirect standardization procedures; for (b) with 15 marks, spend ~30% on deriving the logistic curve from differential equation dP/dt = rP(1-P/L) and discussing properties like inflection point, asymptotic behavior, and symmetry; for (c) with 15 marks, devote remaining ~30% to contrasting TFR, GRR, NRR through formulas, assumptions, and replacement-level interpretations. Structure: brief intro → systematic part-wise treatment → integrated conclusion on demographic measurement evolution.

• Part (a): Lists at least 5 mortality indices (CDR, ASDR, IMR, MMR, U5MR) with formulas; explains purpose of standardization (eliminating age-structure bias for inter-population/temporal comparison); describes direct standardization (applying standard population weights) and indirect standardization (applying standard rates to study population) with step-wise procedure
• Part (a): Cites Indian context—SRS data, Sample Registration System mortality estimates, or NFHS standardized mortality ratios for interstate comparisons
• Part (b): Derives logistic curve by solving dP/dt = rP(1-P/L) with initial condition P(0) = P₀; shows integration steps, substitution, and algebraic manipulation to reach given form with β = (1/r)ln[(L-P₀)/P₀]
• Part (b): Discusses three properties—(i) sigmoid/S-shaped curve with inflection point at P=L/2, t=β; (ii) upper asymptote L (carrying capacity); (iii) growth rate parameter r determining steepness; may add symmetry or point of diminishing returns
• Part (c): Distinguishes TFR (age-specific fertility rates summed, no mortality adjustment, both sexes), GRR (TFR × proportion female births, no mortality, female births only), NRR (GRR adjusted by survival probabilities lₓ to reproductive age, female generation replacement measure); notes NRR=1 indicates exact replacement, TFR≈2.1 replacement level for India
• Part (c): Clarifies that TFR is period measure, GRR/NRR are generation measures; NRR most complete for population projection while TFR most commonly reported

The directive 'explain' demands clear exposition with logical reasoning and supporting evidence. Structure your answer with: (a) ~30% time/space (15 marks) — define WPI, list its three major components (primary articles, fuel & power, manufactured products), then detail the Laspeyres/Fisher methodology for agricultural index numbers with base year selection; (b) ~40% time/space (20 marks) — define G/M/1 assumptions, derive the embedded Markov chain, use generating functions or recursive relations to prove the geometric steady-state distribution πₙ = (1-σ)σⁿ; (c) ~30% time/space (15 marks) — start with life table definitions, express e(x) and e(x+1) in terms of T(x) and l(x), manipulate the identities algebraically to reach the required expression. Conclude each part with a brief summary of significance.

• For (a): WPI definition as base-weighted index measuring wholesale price movements; three commodity groups with their weights (primary articles ~23%, fuel & power ~13%, manufactured products ~64% in India's WPI)
• For (a): Agricultural index methodology — fixed base vs chain base, Laspeyres formula P₀₁ = Σp₁q₀/Σp₀q₀, area and yield indices as relatives with geometric/ arithmetic mean aggregation
• For (b): G/M/1 model specification — general inter-arrival distribution, exponential service (rate μ), single server; embedded Markov chain at arrival epochs with transition probabilities
• For (b): Proof of geometric distribution — derive root σ of generating equation A*(μ(1-z)) = z where σ ∈ (0,1), show πⱼ = (1-σ)σʲ satisfies balance equations and normalization
• For (c): Life table relationships — T(x) = ∫ₓ^ω l(t)dt, e(x) = T(x)/l(x), L(x) = l(x+1) + ½d(x); algebraic manipulation using l(x+1) = l(x)(1-q(x)) to derive the identity
• For (c): Verification that the derived expression satisfies boundary conditions and consistency with complete expectation of life definitions