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Statistics – Optional (Preliminary Examination)
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| Probability
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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.
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| Statistical Methods
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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).
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| Statistical Inference
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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.
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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.
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| Chi-square test, sign test,
Wald-Wolfowitz runs test, run test for randomness, median test, Wilcoxon
test and Wilcoxon-Mann-Whitney test.
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Wal's
sequential probability ratio test, OC and ASN functions, application to
binomial and normal distributions.
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| Loss function, risk function, minimax
and Bayes rules.
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| Sampling Theory and Design of
Experiments
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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.
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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.
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