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

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

Posted at 09 Sep 2026 in UPSC by
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The questions below are from Statistics Optional Paper I 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. (a) $$P\left(\bigcap_{i=1}^n A_i\right) \geq \sum_{i=1}^n P(A_i) – (n-1)$$ Show that for $n$ events $A_1, A_2, \dots, A_n$ $$P\left(\bigcap_{i=1}^n A_i\right) \geq \sum_{i=1}^n P(A_i) – (n-1)$$ $$P(A|A \cup B) = \frac{P(A)}{P(A) + P(B)};\ P(A),\ P(B) > 0$$ For two mutually exclusive events $A$ and $B$ , show that $$P(A|A \cup B) = \frac{P(A)}{P(A) + P(B)};\ P(A),\ P(B) > 0$$ (b) $$p(x, y) = \frac{x^2 + 2y}{80},\ x = 0, 1, 2;\ y = 1, 2, 3, 4$$ (ii) $P(X \leq 1, Y > 2)$ . Let the joint probability mass function of $X$ and $Y$ be $$p(x, y) = \frac{x^2 + 2y}{80},\ x = 0, 1, 2;\ y = 1, 2, 3, 4$$ Find— (i) the marginal probability mass functions of $X$ and $Y$ ; (ii) $P(X \leq 1, Y > 2)$ . 6+4=10 (c) $$f(x)=\frac{e^{-x}x^{\lambda}}{\lambda!};\ x>0$$ $$P[0 < X < 2(\lambda+1)] > \frac{\lambda}{\lambda+1}$$ A random variable $X$ has the probability density function $$f(x)=\frac{e^{-x}x^{\lambda}}{\lambda!};\ x>0$$ where $\lambda$ is a positive integer. Using Chebyshev's inequality, show that $$P[0 < X < 2(\lambda+1)] > \frac{\lambda}{\lambda+1}$$ (d) Describe how Rao-Blackwell theorem enables one to find the UMVUE. 5 $$f(x, \theta)=\theta x^{\theta-1};\ 0 < x < 1,\ \theta > 0$$ Let $X_1, X_2, \dots, X_n$ be a random sample from a population with p.d.f. $$f(x, \theta)=\theta x^{\theta-1};\ 0 < x < 1,\ \theta > 0$$ Show that $\prod_{i=1}^n X_i$ is sufficient statistic of $\theta$. (e) Obtain the likelihood ratio test (LRT) for testing $H_0: \theta \leq \theta_0$ against $H_1: \theta > \theta_0$, using a random sample of size $n$ taken from $P(\theta)$. (1) $X$ is at least one but less than three;
  2. (a) $$f(x, y)=\frac{3}{4}\left(xy+\frac{x^2}{2}\right);\ 0<x<1,\ 0<y<2$$ Let the joint probability density function of $X$ and $Y$ be $$f(x, y)=\frac{3}{4}\left(xy+\frac{x^2}{2}\right);\ 0<x<1,\ 0<y<2$$ Find— – (i) the marginal probability density functions of $X$ and $Y$; – (ii) $E(X)$ and $V(X)$; – (iii) $E(Y)$ and $V(Y)$; – (iv) covariance between $X$ and $Y$; – (v) correlation coefficient between $X$ and $Y$. (b) | — | — | — | — | — | — | — | — | $$\left[ D_{6, 6, 0.05}=\frac{4}{6}\right]$$ The lifetimes, in hours, of two brands of batteries are given below : | Brand A | : | 40 | 30 | 40 | 45 | 55 | 30 | | — | — | — | — | — | — | — | — | | Brand B | : | 50 | 50 | 45 | 55 | 60 | 40 | Examine whether the two brands are different with respect to average life using (i) Kolmogorov-Smirnov test and (ii) median test. $$\left[ D_{6, 6, 0.05} = \frac{4}{6} \right]$$ 8+7=15 (c) Prove that there always exists a sufficient statistic for the one-parameter exponential family of distributions. Using it, show that sample total is sufficient statistic for the parameter for the Poisson model. State whether this estimator is complete.
  3. (a) Let $X_1, X_2, \dots, X_n$ be i.i.d. random variables following $N(\mu, \sigma^2)$. Find the maximum likelihood estimator for $\sigma^2$ and examine whether the same is unbiased. If biased, suggest an unbiased estimator for $\sigma^2$ and examine whether it is consistent. Also compare the mean squared errors of the estimators. (b) A random sample of size $n$ is drawn from $N(\theta, \sigma^2)$. Let the prior distribution of $\theta$ be $N(\mu, \tau^2)$, where $\sigma^2, \mu$ and $\tau^2$ are known. Obtain the Bayes estimator for $\theta$ under (i) squared error loss function and (ii) absolute error loss function. (c) Describe sequential probability ratio test (SPRT). Find the SPRT for testing $H_0: \theta = \theta_0$ against $H_1: \theta = \theta_1$, using random observations from Poisson ($\theta$). Also find the OC function for the test.
  4. (a) Let $Y_1, Y_2, \dots, Y_n$ be $n$ independent observations from a population with mean $\mu$ and variance $\sigma^2$. Obtain the best linear unbiased estimator of $\mu$ and an unbiased quadratic estimator for $\sigma^2$. (b) $$\boldsymbol{\mu}^{(1)} = \begin{pmatrix} 24 \\ 12 \end{pmatrix}, \boldsymbol{\mu}^{(2)} = \begin{pmatrix} 18 \\ 10 \end{pmatrix} \text{ तथा } \sum = \begin{pmatrix} 9 & -5 \\ -5 & 8 \end{pmatrix}$$ Explain the problem of classification in discriminant analysis. In case of two known multivariate normal populations, define Fisher's discriminant function and calculate it for the data given below : $$\boldsymbol{\mu}^{(1)} = \begin{pmatrix} 24 \\ 12 \end{pmatrix}, \boldsymbol{\mu}^{(2)} = \begin{pmatrix} 18 \\ 10 \end{pmatrix} \text{ and } \sum = \begin{pmatrix} 9 & -5 \\ -5 & 8 \end{pmatrix}$$ (c) $$E[\mathbf{Y}'\{I – X(X'X)^{-1}X'\} \mathbf{Y}] = \sigma^2(n – p)$$ Discussing Gauss-Markov linear model $\mathbf{Y}_{n \times 1} = X_{n \times p} \boldsymbol{\beta} + \mathbf{u}_{n \times 1}$ with $E(\mathbf{Y}) = X\boldsymbol{\beta}$ and $D(\mathbf{Y}) =$ variance and covariance matrix of $\mathbf{Y} = \sigma^2 I$, show that $$E[\mathbf{Y}'\{I – X(X'X)^{-1}X'\} \mathbf{Y}] = \sigma^2(n – p)$$ (d) Derive the variance of the sample mean under simple random sample without replacement (SRSWOR). (e) | — | — | — | — | | 1 | 200 | 25 | 4 | | 2 | 300 | 36 | 9 | | 3 | 500 | 49 | 16 | [ P.T.O. A population is divided into three strata. The following data are taken from the population : | Stratum h | N_{h} | S_{h}^{2} | C_{h} | | — | — | — | — | | 1 | 200 | 25 | 4 | | 2 | 300 | 36 | 9 | | 3 | 500 | 49 | 16 | Determine the sample size for each stratum for a total sample of size 100 under (i) proportional allocation and (ii) optimum allocation.
  5. (a) Let $l'\beta$ be any estimable linear function of parameters $\beta_1, \beta_2, \dots, \beta_p$ . Then prove that— (i) there exists a unique linear function $c'Y$ of random variables $Y_1, Y_2, \dots, Y_n$ such that $c \in V(A')$ and $E(c'Y) = l'\beta$ , where $V(A')$ is the vector space generated by row vectors of $A'$ and $E(Y) = A\beta$ ; (ii) variance of $c'Y$ is less than the variance of any other linear unbiased estimator. (b) $$X = \begin{pmatrix} 6 & 9 \\ 10 & 6 \\ 8 & 3 \end{pmatrix}$$ Let $\mathbf{X}_1, \mathbf{X}_2, \dots, \mathbf{X}_N$ be a random sample from $N_p(\mu, \Sigma)$. Show that Hotelling's $T^2$-statistic for testing $H_0: \mu = \mu_0$ against $H_1: \mu \neq \mu_0$ is a generalization of $t$-statistic for testing $H_0: \mu = \mu_0$ against $H_1: \mu \neq \mu_0$ in the univariate case. Calculate $T^2$-statistic for testing $H_0: \mu = \binom{9}{5}$ on the basis of the data taken from a bivariate normal distribution given below : $$X = \begin{pmatrix} 6 & 9 \\ 10 & 6 \\ 8 & 3 \end{pmatrix}$$ (c) Discuss orthogonal polynomial regression on a single variable along with its principal advantages.
  6. (a) $$(i) \quad \mathbf{W}_1 = \frac{1}{4}\mathbf{X}_1 – \frac{1}{4}\mathbf{X}_2 + \frac{1}{4}\mathbf{X}_3 – \frac{1}{4}\mathbf{X}_4$$ $$(ii) \quad \mathbf{W}_2 = \frac{1}{4}\mathbf{X}_1 + \frac{1}{4}\mathbf{X}_2 – \frac{1}{4}\mathbf{X}_3 – \frac{1}{4}\mathbf{X}_4$$ Define multivariate normal distribution and write down its important properties. Let $\mathbf{X}_1, \mathbf{X}_2, \mathbf{X}_3$ and $\mathbf{X}_4$ are mutually independent random vectors following $N_p(\mu, \Sigma)$ . Then find the marginal distributions of $$(i) \quad \mathbf{W}_1 = \frac{1}{4}\mathbf{X}_1 – \frac{1}{4}\mathbf{X}_2 + \frac{1}{4}\mathbf{X}_3 – \frac{1}{4}\mathbf{X}_4$$ $$(ii) \quad \mathbf{W}_2 = \frac{1}{4}\mathbf{X}_1 + \frac{1}{4}\mathbf{X}_2 – \frac{1}{4}\mathbf{X}_3 – \frac{1}{4}\mathbf{X}_4$$ Also obtain the joint distribution of random vectors $\mathbf{W}_1$ and $\mathbf{W}_2$ . (b) $$\pi_i = \frac{nx_i}{\sum x_i}; i = 1, 2, 3, 4$$ A population of $N = 4$ units has values $y_1 = 10, y_2 = 20, y_3 = 30$ and $y_4 = 40$ . Consider a sample of size $n = 2$ without replacement with inclusion probabilities proportional to a size measure $x_i$ , where $x_1 = 1, x_2 = 2, x_3 = 3, x_4 = 4$ . The inclusion probabilities are taken as $$\pi_i = \frac{nx_i}{\sum x_i}; i = 1, 2, 3, 4$$ (i) Suppose the selected sample is $\{2, 4\}$ . Using Horvitz-Thompson estimator, estimate the population total. (ii) Given the joint inclusion probabilities $\pi_{24} = 0.3$ for units 2 and 4, estimate the variance of Horvitz-Thompson estimator and comment on the possibility of negative variance estimate. 7+8=15 (c) Distinguish between complete confounding and partial confounding. Illustrate how you will partially confound a $2^4$-factorial experiment into $2^2$ blocks in 2 replicates with no main effects confounded. State the confounded effects in each replicate. ★★★ SB27—42

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

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