(a) A train is scheduled to arrive at a station at a random time between 09:00 AM and 09:30 AM. The actual arrival time is equally likely to be any moment within this interval.

(i) Define a suitable probability density function (PDF).

(ii) What is

Question

(a) A train is scheduled to arrive at a station at a random time between 09:00 AM and 09:30 AM. The actual arrival time is equally likely to be any moment within this interval.

(i) Define a suitable probability density function (PDF).

(ii) What is the probability that the train will arrive before 09:13 AM ?

(iii) Find the expected arrival time and the variance of the arrival time.

(iv) Given that the train has not arrived by 09:11 AM, what is the expected arrival time ?
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(b) Determine the required sample size for estimating the true weight cereal container from a population of N = 5000.

Based on the past records variance of weight is 121 grams and this estimate should be within 2·5 grams of the true average weight with 99% confidence.

Will there be a change in the size of the sample if we assume population to be infinite ? If so, explain by how much ? (Table enclosed)
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(c) A manufacturing facility specializing in Sheet Metal Production has implemented a Statistical Quality Control Program to monitor and improve process performance. As a part of this initiative, an Engineer has recorded the number of visible surface defects identified on 20 sequential metal sheets, each representing one production unit. The data collected during this inspection phase are given below :

[Table with Sheet No. 1-20 and Defect Counts: 5, 6, 4, 4, 6, 7, 0, 6, 5, 3, 1, 4, 5, 3, 6, 4, 3, 1, 3, 4]

Using the above information :

(i) Determine the Center line (CL), Upper control limit (UCL) and Lower control limit (LCL).

(ii) Plot the appropriate Control Chart and interpret the result.
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(d) A Tech Solutions is a mid-sized IT consulting company specializing in automated HR and payroll systems for small business. The company has successfully digitized most of its operations, relying heavily on its internal computer systems for payroll processing and report generations. However, despite its automation efforts, the computer system has experienced periodic failures. Over the past 20 months, the computer system has broken down as indicated in the table below :

Each breakdown results in an average loss of ₹4500/- due to time delay and technical service expenses. As a preventive measure, company is considering a monthly service contract for preventive maintenance. If they sign up for the contract, the average number of breakdowns is expected to decrease to 1 per month, and the contract would cost ₹3300/- per month.

Based on the given information, advise that the company should go for contract for preventive maintenance or not ?
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(e) Critically analyse the advantages of E Governance for Government, Public and Business giving suitable examples.
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UPSC Answer Check · Accepted Answer

Solve all five numerical and analytical parts systematically, allocating approximately 20% time to each sub-part given equal 10-mark weighting. Begin with precise mathematical derivations for (a) continuous uniform distribution and conditional expectation, (b) finite population correction for sample size, (c) c-chart construction with control limit calculations, (d) expected cost comparison for maintenance decisions, and conclude with (e) critical analysis of e-governance stakeholders. Present calculations step-wise with proper formulae, units, and interpretations.
- (a) Define PDF f(t) = 1/30 for t ∈ [0,30] minutes; calculate P(T<13) = 13/30; derive E(T)=15 min, Var(T)=75; apply memoryless property for conditional expectation E(T|T>11) = 11 + 15 = 26 min or 09:26 AM
- (b) Apply sample size formula n₀ = Z²σ²/E² with Z=2.576, σ²=121, E=2.5; compute n₀ ≈ 139; apply finite population correction n = n₀/(1+n₀/N) ≈ 135; show infinite population increases sample size by ~4 units
- (c) Calculate c̄ = Σcᵢ/k = 80/20 = 4; determine CL=4, UCL=4+3√4=10, LCL=max(0,4-3√4)=0; construct c-chart and interpret no points outside limits, process in statistical control
- (d) Compute expected breakdown cost without contract: compare historical rate (20 breakdowns/20 months = 1/month) vs stated data; calculate expected monthly cost = ₹4500 vs contract cost ₹3300 + ₹4500×1 = ₹7800; recommend against contract if current rate ≤ 1, or detailed cost-benefit if data shows higher breakdowns
- (e) Critically analyse e-governance advantages: Government (efficiency, transparency—MCA21, GSTN), Public (accessibility, grievance redressal—UMANG, Digital India), Business (ease of doing business, reduced corruption—e-way bills, GeM); include challenges like digital divide, cybersecurity

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	25%	12.5	All five sub-parts demonstrate flawless application of correct statistical and management concepts: uniform distribution properties with proper conditional expectation in (a), finite population correction formula in (b), Poisson-based c-chart with exact control limits in (c), expected value decision criterion in (d), and balanced stakeholder analysis in (e); no computational or conceptual errors	Most concepts correctly identified but minor errors in calculation (e.g., wrong Z-value, incorrect variance formula, or missing finite population correction) or incomplete application of conditional probability; e-governance analysis somewhat descriptive rather than critical	Major conceptual misunderstandings such as treating continuous uniform as discrete, ignoring finite population correction entirely, using wrong control chart type (x̄ or p-chart instead of c-chart), or purely descriptive e-governance answer without critical analysis
Framework citation	15%	7.5	Explicitly cites relevant frameworks: continuous uniform distribution X~U(a,b) with proper parameter notation; Cochran's sample size formula with finite population correction; Shewhart's SPC framework with c-chart for attribute data; preventive maintenance decision framework (breakdown vs preventive cost comparison); and e-governance maturity models (UN 4-stage or Gartner's Hype Cycle)	Uses correct formulae but without explicit framework naming; demonstrates awareness of Shewhart control charts and basic decision theory but lacks formal citation of specific models or theorists	No framework identification; presents calculations without context or theoretical grounding; confuses different statistical distributions or quality control methodologies
Case / Indian example	15%	7.5	Integrates relevant Indian examples throughout: Indian Railways punctuality context for (a); FCI or Indian cereal industry sampling for (b); Tata Steel, SAIL or MSME metal fabrication for (c); Indian IT/ITES sector maintenance practices for (d); and specific e-governance initiatives (MCA21, GSTN, UMANG, DigiLocker, GeM, e-Courts, JAM Trinity) with critical evaluation of their impact	Includes some Indian examples but limited to generic mentions like 'Digital India' without specificity; or examples restricted to part (e) while numerical parts lack contextual grounding	No Indian examples; purely theoretical treatment or Western case references; misses opportunity to connect quantitative analysis to Indian business and governance context
Multi-perspective analysis	20%	10	Demonstrates comprehensive multi-perspective analysis: mathematical rigor with interpretation (what do 13/30 probability or 26-minute conditional expectation mean operationally?); sampling efficiency implications for resource-constrained organizations; process control interpretation with actionable insights; cost-benefit analysis from financial and operational risk perspectives; and balanced critical view of e-governance covering efficiency gains alongside digital divide, privacy, and implementation challenges	Solves all parts correctly but treats them as isolated exercises without cross-connecting insights; e-governance analysis covers multiple stakeholders but lacks critical depth on limitations and trade-offs	Fragmented approach with no integration across parts; purely mechanical calculations without interpretation; one-sided e-governance view (only advantages) or stakeholder analysis confined to government perspective alone
Conclusion & recommendation	25%	12.5	Strong, evidence-based conclusions for each part: clear interpretation of arrival time uncertainty for passenger information systems; explicit recommendation on sample size with justification for finite vs infinite population; definitive process control assessment with next steps for improvement; clear maintenance contract decision with sensitivity analysis on breakdown rate assumptions; and nuanced e-governance synthesis recognizing context-specific applicability with forward-looking recommendations	Provides conclusions for most parts but some are merely restatements of calculations rather than insightful recommendations; maintenance decision present but without sensitivity consideration; e-governance conclusion somewhat generic	Missing or weak conclusions; calculations end without interpretation; no clear recommendation on maintenance contract; e-governance analysis trails off without synthesis or fails to address 'critical' aspect of the question directive

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