Теория вероятностей

Чернова Н.И.

Table of contents :
Combinatorics……Page 2
Events and their probabilities……Page 6
Buffon needle……Page 11
Bertrand paradox……Page 12
Sigma-algebra……Page 14
Probability measure……Page 15
Lebesgue Measure……Page 18
Independence……Page 22
Total probability law……Page 24
Bayes’ rule……Page 25
Most probable number of success……Page 26
First success……Page 27
Limit theorem for hypergeometric distribution……Page 28
Polynomial trials……Page 30
Poisson limit theorem……Page 31
Random variables……Page 33
Discrete distributions……Page 34
Distribution function……Page 37
Properties of distribution function……Page 38
Continuous distributions……Page 42
Examples……Page 43
Properties of Gaussian distribution……Page 44
Properties of joint distribution function……Page 47
Classes of joint distributions……Page 48
Independence of random variables……Page 49
Transformations of a single random variable……Page 51
Transformations of two random variables……Page 52
Convolution formula……Page 54
Properties of expectation……Page 56
Variance……Page 58
Properties of variance……Page 60
Expectations and variances of main distributions……Page 61
Covariance……Page 64
Coefficient of correlation……Page 65
Properties of correlation coefficient……Page 66
“A.s.” and “in probability” covergence……Page 70
Chebyshev inequalities……Page 73
LLN (Laws of Large Numbers)……Page 75
Examples……Page 77
Weak convergence……Page 79
CLT (Central Limit Theorem)……Page 81
Moivre – Laplas limit theorem……Page 82
Examples……Page 83
Examples……Page 86
Properties of characteristic functions……Page 87
Proof of CLT……Page 90
Proof of Poisson approximation……Page 92
Appendix: Max & Min……Page 96