Probability and Statistics
(2+2, fourth semester)
Introduction to Probability. Statistical experiment, sample
space. Axioms of Probability. Equally possible outcomes. Geometrical
probability. Statistical determination of probability. Properties
of probability. Applications of combinatorics. Case when the sample
space is infinite.
Conditional probability and independence. Conditional probability.
Total probability formula. Bayes formula. Applications of Bayes formula.
Random variables. Distribution function. Discrete case. Continuous
random variables. Random vectors. Independence of random variables.
Functions of random variables and random vectors.
Numerical characteristics of random variables. Mathematical expectation,
variance. Moments. Covariance and correlation coefficient. Covariance matrix.
Information and entropy.
Characteristic functions. Definiftion and properties. Characteristic
functions of random vectors.
Limit theorems. Various kinds of convergence in Probability theory.
Chebyshev inequality. Laws of large numbers. Central limit theorem.
Convergence of empirical distribution functions.
Conditional distribution. Definition of conditional distribution
with respect to a random variable. Conditional expectation and variance.
Elements of prediction. Applications.
Parameter estimation. Estimation of parameters: mean, variance,
probability. Applications of the Central limit theorem. Confidence intervals.
Testing parameter hypotheses. Hypotheses about the value of a
parameter. Hypotheses about the difference of parameters. T-test. Test for
equality of variances.
Nonparametric testing. Chi squared test. Applications. Kolmogorov-Smirnov
test. Testing independence.
Linear regression. Regression line. General linear regression.
Testing hypotheses about the regression curve.
Monte Carlo methods. Generation of pseudo random numbers. Generation
of random variables with given distributions. Applications.
- Milan Merkle, Petar Vasic: Probability and Statistics -
with applications and examples, Beograd 1995. (In Serbian.)