Indian Statistical Service Exam
1. Probaility: Elements of measure theory,
Classical definitions and axiomatic approach. Sample space. Class of events and
Probability measure. Laws of total and compound probability. Probability of m
events out of n. Conditional probability, Bayes’ theorem. Random variables -
discrete and continuous. Distribution function. Standard probability
distributions - Bernoulli, uniform, binomial, Poisson, geometric, rectangular,
exponential, normal, Cauchy, hypergeometric, multinomial, Laplace, negative
binomial, beta, gamma, lognormal and compound. Poisson distribution. Joint
distributions, conditional distributions, Distributions of functions of random
variables. Convergence in distribution, in probability, with probability one and
in mean square. Moments and cumulants. Mathematical expectation and conditional
expectation. Characteristic function and moment and probability generating
functions Inversion uniqueness and continuity theorems. Borel 0-1 law:
Kolmogorov’s 0-1 law. Tchebycheff’s and Kolmogorov’s inequalities. Laws of large
numbers and central limit theorems for independent variables. Conditional
expectation and Martingales.
2. Statistical Methods: (a) Collection, compilation
and presentation of data, Charts, diagrams and histogram. Frequency
distribution. Measures of location, dispersion, skewness and kurtosis. Bivariate
and multivariate data. Association and contingency. Curve fitting and orthogonal
polynomials. Bivariate normal distribution. regression-linear, polynomial.
Distribution of the correlation coefficient, Partial and multiple correlation,
Intraclass correlation, Correlation ratio.(b) Standard errors and large sample
test. Sampling distributions of x,s2, t, chisqure and F; tests of significance
based on them, Small sample tests. (c) Non-parametric tests-Goodness of fit,
sign, median, run, Wicloxon, Mann-Whitney, Wald-Wolfowitz and Kolmogorov-Smirnov.
Rank order statistics-minimum, maximum, range and median. Concept of Asymptotic
relative effciency. 3. Numerical Analysis Interpolation formulae (with remainder
terms) due to Lagrange, Newton-Gregory, Newton Divided different, Gauss and
Striling. Euler-Maclaurin’s summation formula. Inverse interpolation. Numerical
integration and differentiation. Difference equations of the first order. Linear
difference equations with constant coefficients.