57 questions from the UPSC 2020 examination.
57 questions
Let \{X_n\} be a sequence of independent random variables such that P(X_k = \frac{k}{n}) = \frac{1}{n}, k = 1, 2, ..., n Show that X_n converges in law to a uniform distribution. State the result you have used.
Obtain characteristic function of uniform distribution U(0, 1).
The lifetime in hours of each bulb manufactured by a particular company follows an independent exponential distribution with mean \theta. To test the hypothesis H_0 : \theta = 2000 versus H_1 : \theta = 1000, an experiment sets up an experiment with 50 bulbs with 5 bulbs in each of 10 different locations to examine their lifetimes. To get quick preliminary results, the experimenter decides to stop the experiment as soon as one bulb fails at each location. Let Y_i denote the lifetime of the first bulb to fail at location i, i = 1, 2, ..., 10. Obtain the most powerful test of size 0.05 to test H_0 versus H_1 based on the available lifetimes. If the available data is 510, 9752, 5650, 12385, 230, 4225, 860, 300, 3000 and 1500, give your conclusion. [Given that : P(\chi_{(20)}^{2} < 10.851) = 0.05]
Define Hotelling's-T^2. Show that T^2 statistic is a function of likelihood ratio criteria.
Two types of tests viz : written and oral were conducted for seeking admission to MBA course. The result of the test given by the students are as follows: x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \begin{matrix} \text{marks obtained in written test} \\ \text{marks obtained in oral test} \end{matrix} Let \pi_1 : students eligible for admission to MBA course \pi_2 : students not eligible for admission to MBA course \mu_1 = \begin{pmatrix} 30.6 \\ 14.4 \end{pmatrix}, \Sigma_1 = \begin{pmatrix} 5.60 & -0.62 \\ -0.62 & 2.08 \end{pmatrix} \mu_2 = \begin{pmatrix} 24.8 \\ 11.2 \end{pmatrix}, \Sigma_2 = \begin{pmatrix} 5.60 & -0.62 \\ -0.62 & 2.08 \end{pmatrix} If one student gets 29.0 marks in written test and 12.0 in oral test, classify the student as eligible or not based on Fisher's linear discriminant function. Also find the probability of misclassification. [Given that P(0 < Z < 1.8277) = 0.4664]
What are control charts for attributes? Discuss p and c charts. The following data refer to number of defectives in 10 samples each of size 100 items. Construct an appropriate control chart and interpret the control limits : Sample No. : 1 2 3 4 5 6 7 8 9 10 No. of Defectives : 4 8 11 3 11 7 7 16 12 6
Differentiate between a complete life table and an abridged life table. Explain the terms involved in life table. In a sample survey of a locality, the number of males of ages 45 and 46 were 30450 and 30320 respectively. Calculate q45.
Let X_1, X_2, ..., X_n be a random sample from uniform distribution U(\theta - \frac{1}{2}, \theta + \frac{1}{2}). Show that Y = (X_{(1)}, X_{(n)}) is sufficient statistic for \theta but not complete, where X_{(1)} = \min (X_1, X_2, ..., X_n) X_{(n)} = \max (X_1, X_2, ..., X_n)
Show that in pps sampling, with replacement, an unbiased estimator of the population mean \bar{Y} is \bar{y}_{pps} = \frac{1}{n} \sum_{i=1}^{n} \frac{y_i}{Np_i} = \frac{1}{n} \sum_{i=1}^{n} Z_i = \bar{Z}, with variance V(\bar{y}_{pps}) = \frac{1}{n} \sum_{i=1}^{N} p_i \left(\frac{y_i}{Np_i} - \bar{Y}\right)^2 where p_i = \frac{X_i}{X} be the probability that the i^{th} unit is selected in a sample, such that \sum p_i = 1, Z_i = \frac{y_i}{Np_i}.
Let X ~ uniform U(0, 2\theta), \theta > 0 distribution. Define Y = \max(X, 2\theta - X). Find E(Y).
Discuss variate difference method with an application.
State inversion theorem on characteristic function. Hence, find the density function corresponding to the characteristic function \phi(t) = \begin{cases} 1 - |t|, & |t| <= 1 \\ 0, & |t| > 1 \end{cases}
For a group of policies, losses follow the distribution function F(x) = 1 - \frac{\theta}{x}, 0 < x < \infty. A sample of 20 losses resulted in the following: Interval: x <= 10, Number of losses: 9 Interval: 10 < x <= 25, Number of losses: 6 Interval: x > 25, Number of losses: 5 (i) Obtain maximum likelihood estimator (MLE) of \theta. (ii) Find Cramer-Rao Lower Bound (CRLB) using MLE of \theta. (iii) Using CRLB and asymptotic distribution of MLE, obtain 95% confidence interval for \theta.
What are sampling plans? Define consumer's risk, producer's risk and OC curve. In a double sampling plan with N = 1000, n1 = 50, c1 = 3, n2 = 100 and c2 = 7, explain how you would draw your conclusion.
Explain logistic curve for population projection. Describe any one method for fitting logistic curve to the population data.
What do you understand by scaling of test items? What are raw scores, standard scores and T scores? Explain. In a behavioural study on a group of students with mean 86 and standard deviation 15, Ram scored 91 and Shyam scored 83. Express these raw scores as standard scores with mean 500 and standard deviation 100.
Explain the concept of validity of a test. Describe various types of validity.
A random sample of 14 male students from a rural Junior High School and an independent random sample of 16 male students from an urban Junior High School were given a test to measure their level of mental health. The obtained data are given in the following table: Mental Health Scores of Junior High School's male students: Rural : 28, 49, 42, 30, 33, 36, 49, 31, 41, 46, 41, 21, 20, 28 Urban : 37, 22, 24, 24, 27, 44, 34, 26, 20, 26, 25, 45, 32, 40, 45, 25 Apply Median Test to test the hypothesis at 5% level of significance that their median mental health level scores are equal or not. [Given that \chi_{(1, 0.05)}^{2} = 3.841]
Define reliability of a system with an example along with the criteria affecting it.
What is systematic sampling? In usual notation, prove that V(\bar{y}_{sys}) = \frac{nk - 1}{nk} \frac{S^2}{n} (1 + (n - 1) \rho), where \rho is the intra-class correlation coefficient between the units of the same systematic samples. Hence, determine the minimum value of \rho for V(\bar{y}_{sys}) to be non-negative.
Explain in brief how you will determine orders of autoregressive and moving average terms while fitting time series using ARIMA modelling.
Write a note on the characteristics of game theory and discuss its limitations.
Discuss Simple Lattice Design. Let \beta_{1}, ..., \beta_{p} denote row block effects and \beta'_{.1}, ..., \beta'_{.p} denote column block effects and \zeta_{ij} denote the effect of (i, j)^{th} treatment, where number of treatments t = p^2. How do you test the hypothesis H_0 : \zeta_{11} = \zeta_{12} = ... = \zeta_{pp} = \tau using ANOVA table?
Discuss Markov chain and transition probability matrix with an example. For the following transition probability matrix, determine the market share of brands A and B from period 0 to 1, when their initial market share breakdown is 50% : From To A B A 0.9 0.1 B 0.5 0.5
Let Y_1, Y_2, Y_3 be uncorrelated variables with common variance \sigma^2 such that E(Y_1) = \beta_1 + \beta_2 E(Y_2) = 2\beta_1 E(Y_3) = \beta_1 - \beta_2 Compute residual mean square.
Let X_1 and X_2 be two iid random variables with normal N(\theta, 1) distribution. Further, consider Bernoulli random variable V with P(V = 1) = 1/4 and which is independent of X_1 and X_2. Define X_3 as X_3 = \begin{cases} X_1, \text{ if } V = 0 \\ X_2, \text{ if } V = 1 \end{cases} For testing the hypothesis H_0 : \theta = 0 versus H_1 : \theta = 1, consider the test Reject H_0 if \frac{X_1 + X_2 + X_3}{3} > C Find C such that the test size becomes 0.05. Also compute power of the test. [Given that : P(Z > 1.64) = 0.05, P(|Z| > 1.96) = 0.05, P(Z > 0.30) = 0.03821]
The random variable X takes values on the interval [0, 2]. Given that P(X = 1) = 0.25, F(x|x < 1) = x^2, F(x|x > 1) = x - 1, E(X) = 1. Find P(X <= 1).
Differentiate between process control and product control with examples.
Let n = 4, p = 2, X = \begin{pmatrix} X_1 \\ X_2 \end{pmatrix}, where X_1 : weight (in kg), X_2 : height (in cm) of infant. Sample data matrix X = \begin{pmatrix} 30 & 40 & 30 & 30 \\ 40 & 40 & 30 & 35 \end{pmatrix} Test at 5% level of significance the null hypothesis H_0 : \mu = \mu_0 = \begin{pmatrix} 3.0 \\ 35.0 \end{pmatrix} [Given that : F_{((2, 2), 0.05)} = 19.00]
A student fits the following linear regression model to a given data: y_i = \alpha + \beta x_i + u_i, i = 1, 2, ..., n u_i's are iid normal values (N(0, \sigma^2)), where \sigma^2 is unknown. The following information is available: n = 7, \sum(x_i - \bar{x})^2 = 280000, \sum(x_i - \bar{x})(y_i - \bar{y}) = 16500 95% confidence interval for \beta is : (0.030, 0.088). Calculate what proportion of the total variability of the responses is explained by the model. Given that t_{0.025, 5} = 2.57.
An insurer offers a health plan to the employees of a large company. As part of this plan, individually any employee may choose exactly two or none of the supplementary coverages A, B and C. The proportions of the company employees that choose coverage A, B and C are 1/6, 1/2 and 7/12 respectively. Determine the probability that a randomly chosen employee will choose no supplementary coverage.
Let X ~ N_2(\mu, \Sigma) with probability distribution function (pdf), f(x) = C e^{-Q/2}, where Q = 3x^2 + 2y^2 - 2xy - 32x + 4y + 92. Obtain \mu and \Sigma.
Explain the general Linear Programming Problem. Solve the following LPP by Charnes' Big-M method : Minimize Z = 2x1 + x2 subject to the constraints 3x1 + x2 = 3, 4x1 + 3x2 >= 6, x1 + 2x2 <= 3, x1 >= 0, x2 >= 0
Define two-person zero-sum game and saddle point of a payoff matrix. Solve the following game : Player B Player A 1 2 3 4 5 I 3 5 4 9 6 II 5 6 3 7 8 III 8 7 9 8 7 IV 4 2 8 5 3
Solve the following Transportation Problem by MODI method : Destination Origin D1 D2 D3 D4 Availability O1 1 2 1 4 30 O2 3 3 2 1 50 O3 4 2 5 9 20 Demand 20 40 30 10
Obtain reliability and hazard functions of exponential and lognormal distributions.
Explain the concept of multicollinearity in general linear model. How will you detect it? Discuss its impact on OLS estimators and their variances.
Define Gauss-Markov linear model. State and prove necessary and sufficient condition for the linear function l^{\prime}\beta of the parameters to be linearly estimable.
Define Crude Death Rate. Write its merits and demerits. Generally, what is the range of Crude Death Rate? Is there any relation between female CDR and male CDR? If yes, then state the relation.
Describe BIBD and show that b >= v, where symbols have usual meanings.
Define time series. Distinguish between stationary and non-stationary time series. Give an example of stationary and non-stationary time series.
A population consists of 6 units with values 6, 3, 1, 4, 2 and 5. Write down all possible samples of size 2 without replacement and verify that V(\bar{y}) = \frac{N-n}{N.n} S^2.
Define the concept of duality in Linear Programming Problem (LPP). Write down the dual of the following LPP : Maximize Z = 2x1 + 5x2 + 6x3 subject to the constraints 5x1 + 6x2 - x3 <= 3, -2x1 + x2 + 4x3 <= 4, x1 - 5x2 + 3x3 <= 1, -3x1 - 3x2 + 7x3 <= 6, x1, x2, x3 >= 0
Discuss in detail Gompertz curve for population growth and explain the method of its fitting.
What do you mean by Partial Confounding ? Describe a 2^3-factorial design arranged in 2 blocks with 4 replications, where ABC, BC, AC and AB are confounded. Also prepare ANOVA table for such design.
What is Dickey-Fuller (DF) test? Describe how you will use this test for testing stationarity of any given time series.
What is the problem of identification? Explain it with the help of an example. Establish rank and order conditions of identifiability.
For a geometric distribution p(x) = 2^{-x}; x = 1, 2, 3, ..., show that Chebyshev's inequality gives P(|X - 2| <= 2) > 1/2, while the actual probability is 15/16.
Let \theta > 0 be the unknown parameter and X_1, X_2, ..., X_n be a random sample from the distribution with the probability distribution function (pdf) f(x, \theta) = \begin{cases} \frac{2x}{\theta^2}, & 0 <= x <= \theta \\ 0, & \text{otherwise} \end{cases} Find maximum likelihood estimator (MLE) of \theta and its mean squared error (MSE).
To test sequentially the hypothesis H_0 for which the distribution is given by P(X = -1) = P(X = 1) = P(X = 2) = 1/3 against the alternative H_1 that is given by P(X = -1) = P(X = 1) = 1/4, P(X = 2) = 1/2 it is decided to continue sampling as long as -\left(\frac{n+1}{2}\right) < S_n < \left(\frac{n+2}{2}\right) where S_n = \sum_{i=1}^{n} X_i, and X_i's are successive observations. Compute the probability under H_0 and H_1 that the procedure will terminate with the 2^{nd} observation or earlier.
Let R be the ratio of population totals of 'y' and 'x' and \hat{R} is the estimate of R. Obtain V(\hat{R}). Also find the estimate \hat{V}(\hat{R}).
Let X and Y be iid random variables with P(X = k) = 2^{-k}, k = 1, 2, 3, ... . Find P(X > Y).
What are the main functions of Central Statistics Office (CSO) and Directorate of Economics and Statistics established in different States? Explain in detail.
Discuss Assignment Problem. Consider the problem of assigning five jobs to five persons. The assignment costs are given as follows : Job Person 1 2 3 4 5 A 8 4 2 6 1 B 0 9 5 5 4 C 3 8 9 2 6 D 4 3 1 0 3 E 9 5 8 9 5 Determine the optimum assignment schedule.
What do you understand by queuing theory? What are its advantages? Discuss the characteristics of M/M/1 model.
Calculate gross and net reproduction rates from the following data considering sex ratio at birth to be 105 males to 100 females : Age (in years) Age-Specific Fertility Rate Female Life Table Stationary Population 15-19 0.0696 4180 20-24 0.2346 4123 25-29 0.1897 4063 30-34 0.1143 4001 35-39 0.0611 3934 40-44 0.0285 3860 45-49 0.0101 3763
Derive failure rate of a device assuming normal distribution. Suppose the life of an equipment is known to be normally distributed with mean 100 hours and standard deviation 20 hours. What is the probability that the equipment would last at least 130 hours? (Normal Distribution Table is given in Page No. 9)