Part A - Preliminary Examination - Optional Subjects
Examination Syllabus - Subject : Statistics
Random experiment, sample space, event, algebra of events, probability on a
discrete sample space, basic theorems of probability and simple examples based
there on, conditional probability of an event, independent events, Bayes'
theorem and its application, discrete and continuous random variables and their
distributions, expectation, moments, moment generating function, joint
distribution of two or more random variables, marginal and conditional
distributions, independence of random variables, covariance, correlation,
coefficient, distribution of function of random variables. Bernoulli, binomial,
geometric, negative binomial, hypergeometric, Poisson, multinomial, uniform,
beta, exponential, gamma, Cauchy, normal, longnormal and bivariate normal
distributions, real-life situations where these distributions provide
appropriate models, Chebyshev's inequality, weak law of large numbers and
central limit theorem for independent and identically distributed random
variables with finite variance and their simple applications.
Concept of a statistical population and a sample, types of data, presentation
and summarization of data, measures of central tendency, dispersion, skewness
and kurtosis, measures of association and contingency, correlation, rank
correlation, intraclass correlation, correlation ratio, simple and multiple
linear regression, multiple and partial correlations (involving three variables
only), curve-fitting and principle of least squares, concepts of random sample,
parameter and statistic, Z, X2, t and F statistics and their properties and
applications, distributions of sample range and median (for continuous
distributions only), censored sampling (concept and illustrations).
Unbiasedness, consistency, efficiency, sufficiency, Completeness, minimum
variance unbiased estimation, Rao-Blackwell theorem, Lehmann-Scheffe theorem,
Cramer-Rao inequality and minimum variance bound estimator, moments, maximum
likelihood, least squares and minimum chisquare methods of estimation,
properties of maximum likelihood and other estimators, idea of a random
interval, confidence intervals for the paramters of standard distributions,
shortest confidence intervals, large-sample confidence intervals.
Simple and composite hypotheses, two kinds of errors, level of significance,
size and power of a test, desirable properties of a good test, most powerful
test, Neyman-Pearson lemma and its use in simple example, uniformly most
powerful test, likelihood ratio test and its properties and applications.
Chi-square test, sign test, Wald-Wolfowitz runs test, run test for
randomness, median test, Wilcoxon test and Wilcoxon-Mann-Whitney test.
Wal's sequential probability ratio test, OC and ASN functions, application to
binomial and normal distributions.
Loss function, risk function, minimax and Bayes rules.
Sampling Theory and Design of Experiments
Complete enumeration vs. sampling, need for sampling, basic concepts in
sampling, designing large-scale sample surveys, sampling and non-sampling
errors, simple random sampling, properties of a good estimator, estimation of
sample size, stratified random sampling, systematic sampling, cluster sampling,
ratio and regression methods of estimaton under simple and stratified random
sampling, double sampling for ratio and regression methods of estimation,
two-stage sampling with equal-size first-stage units.
Analysis of variance with equal number of observations per cell in one, two
and three-way classifications, analysis of covariance in one and two-way
classifications, basic priniciples of experimental designs, completely
randomized design, randomized block design, latin square design, missing plot
technique, 2n factorial design, total and partial confounding, 32 factorial
experiments, split-plot design and balanced incomplete block design.
Courtesy : UPSC