# Theory of Probability and Mathematical Statistics

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Al-Farabi Kazakh National University

Faculty of mechanics and mathematics

"APPROVED"

By scientific council

Of faculty of mechanics and mathematics

"__"_______________2015 y.

Protocol №_____

Chairman of the Scientific Council

_________А.B.Kydyrbekuly

Program SEB

on fundamental disciplines for bachelor
specialty - Information systems

Almaty 2015

Mathematical analysis

1. Convergent sequences and their properties. Criterion of convergence of Cauchy sequences. Yield functions. Cauchy criterion on existence of the limit functions.

2. Continuity of functions. Properties of continuous functions on the segment.
The points of discontinuity of the functions and their classification.

3. Fundamental theorems on differentiable functions. Taylor's formula for functions of one variable

4. A necessary and sufficient condition for the integrability of functions. Classes of integrable functions. Theorem on the mean value of the definite integral.

5. Improper integrals of the first and second kind. Sufficient convergence of the improper integral.

6. The functions of several variables. Limit of a function of many variables. Taylor's formula for functions of several variables.

7. Local extrema of function of many variables. The necessary and sufficient conditions for a local extremum. Constrained optimization.

8. Number series. Absolutely and conditionally convergent series. Sufficient conditions for the convergence of numerical series.

9. Functional sequences and series. Sufficient uniform convergence of functional sequences and series.

10. Power series and their convergence region. Term by term integration and differentiation of power series term by term. Expansion of functions in power series.

11. The basic properties of the double integral. Change of variables in double and triple integral

12. The curvilinear integrals of the first and second series.

Algebra and geometry

Algebraic form of complex numbers, operations on them and their properties. Designation on the plane and trigonometric form of a complex number. Formula Moivre. The formula for calculating the root of the n-th degree of the complex number.

2. The axioms of a vector space. The linear dependence and independence of vectors. Properties of linear dependence.

3. Divisibility properties of polynomials. The greatest common divisor of polynomials. Euclid's algorithm for finding the greatest common divisor.

4. The inverse matrix. The criterion of reversibility of the matrix.

5. The vector and mixed product of vectors and their geometric meaning.

6. The equation of a line on the plane. Distance from point to line. The angle between the straight lines on the plane.

7. Equations plane in space. The distance from the point to the plane. The angle between planes.

8. Canonical equations of second-order curves. The eccentricity of the ellipse and the headmistress and hyperbole.

Theory of Probability and Mathematical Statistics

1. Elements of combinatorics. Various selection circuit balls from an urn.

2. . Conditional probability. The formula multiplication of probabilities.

3. The formula of total probability. Bayes' formula.

4. Independent events. Examples.

5. The Bernoulli scheme. Formula Bernoulli. De Moivre-Laplassa, folrmula Poisson approximation.

6. The random variables. The law of distribution and the distribution function of the random variable. Discrete and continuous random variables.

7. Math Expectation and variance of the random variable. Properties.

8. Covariance. Correlation coefficient. Properties.

9. The law of large numbers. The central limit theorem.

10. Empirical distribution function. The sample mean and sample variance.

11. Estimates. Classification of estimates (unbiasedness, consistency, efficiency

12.Confidence intervals for the parameters of the normal distribution.

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