UPSC Civil Services (Main) Examination 2026 — Statistics Optional Paper II: Question Paper | Plutus IAS

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09 Sep UPSC Civil Services (Main) Examination 2026 — Statistics Optional Paper II: Question Paper | Plutus IAS

Posted at 09 Sep 2026 in UPSC by
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The questions below are from Statistics Optional Paper II of UPSC Civil Services (Main) Examination 2026 (held 2026-08-30) — the actual paper, which is public government content. This page carries the questions for reference and revision; model answers for this paper are not published here.

Official paper (PDF): upsc.gov.in.

  1. $$f(\theta) = \frac{\alpha}{\lambda} \left( \frac{\theta}{\lambda} \right)^{\alpha-1} \exp\left( -\frac{\theta}{\lambda} \right)^\alpha, \quad 0 \leq \theta < \infty$$ Suppose that a given individual in a population has a survival time which is exponential with a hazard rate $\theta$. Each individual's hazard rate $\theta$ is potentially different and is sampled from a Weibull distribution with probability density function : $$f(\theta) = \frac{\alpha}{\lambda} \left( \frac{\theta}{\lambda} \right)^{\alpha-1} \exp\left( -\frac{\theta}{\lambda} \right)^\alpha, \quad 0 \leq \theta < \infty$$ Let $X$ be the life length of a randomly chosen individual of this population. – (i) Find the survival function of $X$. – (ii) Find the hazard rate of $X$. What is the shape of the hazard rate ? (b) $$A.Q.L. = 0.05$$ $$LTPD = 0.20$$ Construct a single sampling plan for attributes, given the following data : $$A.Q.L. = 0.05$$ $$\text{Producer's risk} = 0.05$$ $$\text{Consumer's risk} = 0.10$$ $$LTPD = 0.20$$ (c) Describe the effect of sample size on $\bar{X}$, $R$ and $s$ charts for the sample sizes $n \leq 5$, $5 < n \leq 10$ and $n > 10$. (d) $$P = \begin{pmatrix} 0.6 & 0.3 & 0.1 \\ 0.2 & 0.6 & 0.2 \\ 0.4 & 0.4 & 0.2 \end{pmatrix}$$ Given a transition probability matrix with three states {1, 2, 3} as : $$P = \begin{pmatrix} 0.6 & 0.3 & 0.1 \\ 0.2 & 0.6 & 0.2 \\ 0.4 & 0.4 & 0.2 \end{pmatrix}$$ Find the value of $P(X_5 = 2 | X_3 = 3)$ and $P(X_3 = 1 | X_0 = 3)$ . (e) What are the key features of SPSS software? How does it compare to R software?
  2. How will you draw the OC curve for a single sampling plan with sample size 5, if the acceptance number is $c = 2$ , assuming the lot size to be large? Explain. (b) $$6, 2, 5, 1, 2, 2, 3, 5, 3, 4, 12, 4, 4, 1, 3, 5, 4, 1, 4, 3, 5, 4, 2, 3.$$ Each day, a sample of 50 items from a production process was examined. The number of defectives found in each sample was as follows : $$6, 2, 5, 1, 2, 2, 3, 5, 3, 4, 12, 4, 4, 1, 3, 5, 4, 1, 4, 3, 5, 4, 2, 3.$$ Draw a suitable control chart and check for control. What control limits would you suggest for subsequent use? (c) $$x_1 – 2x_2 + x_3 \geq 4$$ $$2x_1 + x_2 + x_3 \leq 8$$ $$x_1 – x_3 \geq 0$$ $$\text{और } x_1 \geq 0, x_2 \geq 0, x_3 \geq 0$$ (i) Distinguish between slack variables and surplus variables. Discuss how they are used in a linear programming problem. (ii) Use the dual simplex method to solve the following Linear Programming Problem (LPP) : $$\text{Minimize } Z = 2x_1 + x_2 + 3x_3$$ subject to $$x_1 – 2x_2 + x_3 \geq 4$$ $$2x_1 + x_2 + x_3 \leq 8$$ $$x_1 – x_3 \geq 0$$ $$\text{and } x_1 \geq 0, x_2 \geq 0, x_3 \geq 0$$
  3. | — | — | — | — | — | — | | P_{1} | 2 | 3 | 11 | 7 | 6 | | P_{2} | 1 | 0 | 6 | 1 | 1 | | P_{3} | 5 | 8 | 15 | 9 | 10 | (i) What is a Transportation problem? Give a general mathematical model of a transportation problem. 5(ii) Use Vogel's approximation method to solve the following transportation problem : 10 Distribution Centre | Plant | D_{1} | D_{2} | D_{3} | D_{4} | Capacity | | — | — | — | — | — | — | | P_{1} | 2 | 3 | 11 | 7 | 6 | | P_{2} | 1 | 0 | 6 | 1 | 1 | | P_{3} | 5 | 8 | 15 | 9 | 10 | | Demand | 7 | 5 | 3 | 2 | | (c) (i) Define a Poisson process and state at least three applications of the same. (ii) In a local post office, a particular service is operated by one person. The arrival pattern of customers on a particular day follows a Poisson distribution with a mean arrival rate of 10 people per hour. Customers are served on a First-Come, First-Served (FCFS) basis and the service time is estimated to be exponentially distributed with an average time of 4 minutes. Determine : (I) Probability (there is no queue) (II) The average size of the queue (III) The expected waiting time in the queue (IV) Probability (a customer will spend less than 12 minutes in the queue)
  4. (a) $$C_t = \alpha + \beta Y_t + u_t$$ $$Y_t = C_t + Z_t$$ Explain the 2-stage least squares (2SLS) method of estimation for the following simultaneous equation systems : $$C_t = \alpha + \beta Y_t + u_t$$ $$Y_t = C_t + Z_t$$ where C = Aggregate consumption expenditure Y = National income Z = Non-consumption expenditure $u_t$ = A stochastic variable (b) Discuss methods of detecting the presence of heteroscedasticity. Explain the Goldfeld-Quandt test for detecting it, stating the assumptions required to apply this test. (c) $$Y = X\beta + u$$ जहाँ $u \sim N(0, \sigma^2 I_n)$ For the General Linear Model $$\mathbf{Y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{u}$$ where $\mathbf{u} \sim \mathbf{N}(\mathbf{0}, \sigma^2 I_n)$ obtain the Maximum Likelihood Estimator (MLE) $\hat{\boldsymbol{\beta}}$ of $\boldsymbol{\beta}$. Further, let $\mathbf{e} = (\mathbf{Y} – \mathbf{X}\hat{\boldsymbol{\beta}})$, and $\mathbf{X}$ is a matrix of order $n \times k$ with rank of $\mathbf{X}$, $k < n$. If $\mathbf{e}'\mathbf{e} = \mathbf{u}'\mathbf{M}\mathbf{u}$, where $\mathbf{M} = (I – \mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}')$, show that $\frac{\mathbf{e}'\mathbf{e}}{\sigma^2}$ follows chi-square distribution with $(n – k)$ degrees of freedom (d.f.). 10 (d) What are the usual sources of data on vital events ? What types of error usually occur in census data on age ? How are such errors adjusted ? 10 (e) | — | — | — | — | — | — | — | — | — | | f : | 1 | 4 | 6 | 10 | 8 | 13 | 18 | 2 | Why is it considered desirable to convert raw scores to standard scores ? Below is given the distribution of test scores of 62 children : | Test Score : | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | | — | — | — | — | — | — | — | — | — | | f : | 1 | 4 | 6 | 10 | 8 | 13 | 18 | 2 | Obtain standard scores with mean 50 and standard deviation 10.
  5. Define autocorrelation of order k and correlogram. For an infinite series generated by the moving average of random components with equal weights, obtain the correlogram of a moving average of extent m. (b) | — | — | — | — | — | | | p_{1} | q_{1} | p_{2} | q_{2} | | A | 10 | 2 | 15 | 1 | | B | 15 | 3 | 10 | 3 | | C | 20 | 4 | 15 | 4 | Explain the uses and limitations of Index Numbers. Calculate Laspeyres', Paasche's and Fisher's Indices for the following data : | Commodity | Time Period I | | Time Period II | | | — | — | — | — | — | | | p_{1} | q_{1} | p_{2} | q_{2} | | A | 10 | 2 | 15 | 1 | | B | 15 | 3 | 10 | 3 | | C | 20 | 4 | 15 | 4 | Also, examine whether Paasche's and Fisher Index Numbers satisfy (i) Time Reversal Test, and (ii) Factor Reversal Test or not. (c) (i) Define Morbidity and discuss the different measures of morbidity. (ii) Explain stable population. Prove that the proportion of population at any age 'x' and time 't', denoted by C(x, t) is independent of time under a stable population, i.e., C(x, t) = C(x).
  6. (i) Define a life table and mention its uses. (ii) The number of persons dying at age 75 is 476, and the complete expectations of life at ages 75 and 76 years are 3.92 and 3.66 years, respectively. Find the number of persons living at ages 75 and 76 years. (b) | — | — | — | | 15 – 19 | 16.0 | 260 | | 20 – 24 | 16.4 | 2244 | | 25 – 29 | 15.8 | 1894 | | 30 – 34 | 15.2 | 1320 | | 35 – 39 | 14.8 | 916 | | 40 – 44 | 15.0 | 280 | | 45 – 49 | 14.5 | 145 | (i) Compute the Age-specific Fertility Rate, the Total Fertility Rate, and the Gross Reproduction Rate, from the data given below : | Age group of child-bearing females | No. of women ('000) | Total Births | | — | — | — | | 15 – 19 | 16.0 | 260 | | 20 – 24 | 16.4 | 2244 | | 25 – 29 | 15.8 | 1894 | | 30 – 34 | 15.2 | 1320 | | 35 – 39 | 14.8 | 916 | | 40 – 44 | 15.0 | 280 | | 45 – 49 | 14.5 | 145 | Assume that the proportion of female births is 46.2 percent. (ii) Define “Reproduction Rates”, and explain how far they may be looked upon as indices of population growth. (c) For the general linear model $$Y_{n \times 1} = X_{n \times k} \beta_{k \times 1} + u_{n \times 1}$$ with $$E(u) = 0$$ and $$E(uu') = \sigma^2 I_n$$, where $$X_{n \times k}$$ is a matrix of fixed numbers with rank $$k$$ ($$k < n$$), obtain the ordinary least squares estimator $$\hat{\beta}$$ of $$\beta$$ and the dispersion matrix of $$\hat{\beta}$$. Also, prove that $$\hat{\beta}$$ is the best linear unbiased estimator (BLUE) of $$\beta$$.
  7. $$Y_1 = \alpha_{10} + \alpha_{12}Y_2 + \alpha_{13}Y_3 + \beta_{11}X_1 + u_1$$ $$Y_2 = \alpha_{20} + \alpha_{23}Y_3 + \beta_{21}X_1 + \beta_{22}X_2 + u_2$$ $$Y_3 = \alpha_{30} + \alpha_{31}Y_1 + \beta_{31}X_1 + \beta_{32}X_2 + u_3$$ $$Y_4 = \alpha_{40} + \alpha_{41}Y_1 + \alpha_{42}Y_2 + \beta_{43}X_3 + u_4$$ Consider the following structural equation model assuming Y's as endogenous and X's as pre-determined variables : $$Y_1 = \alpha_{10} + \alpha_{12}Y_2 + \alpha_{13}Y_3 + \beta_{11}X_1 + u_1$$ $$Y_2 = \alpha_{20} + \alpha_{23}Y_3 + \beta_{21}X_1 + \beta_{22}X_2 + u_2$$ $$Y_3 = \alpha_{30} + \alpha_{31}Y_1 + \beta_{31}X_1 + \beta_{32}X_2 + u_3$$ $$Y_4 = \alpha_{40} + \alpha_{41}Y_1 + \alpha_{42}Y_2 + \beta_{43}X_3 + u_4$$ where $$u_1, u_2, u_3$$ and $$u_4$$ are stochastic terms. Use rank and order conditions to identify the equations. (b) x : 91 92 93 94 95 96 97 98 99 100 101 $d_x$ : 296 209 144 93 58 34 18 10 5 3 1 Given $l_{91} = 871$ and : x : 91 92 93 94 95 96 97 98 99 100 101 $d_x$ : 296 209 144 93 58 34 18 10 5 3 1 where $l_x$ and $d_x$ have their usual meanings as in a life table, find the probability that : – (i) a person aged 93 will die within three years. – (ii) a person aged 92 will survive up to age 96. – (iii) three persons aged 92, 93 and 94 will survive 4 years. – (iv) of three persons aged 93, 94 and 95, exactly two of the three will be alive in 4 years. – (v) all the three persons aged 93, 94 and 95 will die in 4 years. (c) (i) Explain the use of parallel tests in psychological studies. 5 (ii) Explain how you will combine the ranks of a number of subjects given by several judges. 10

Questions reproduced from the official paper for study purposes; verify on the exam-conducting body’s official website.

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