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A Table of Contents A TABLE OF CONTENTS LECTURE 1 ....................................................................................................................................................................... 5 1.1. Random variables. Probability distribution law .................................................................................................. 5 1.2. Cumulative distribution function .......................................................................................................................... 6 1.3. Distribution set ..................................................................................................................................................... 7 1.4. Probability density function ................................................................................................................................. 8 LECTURE 2 ..................................................................................................................................................................... 10 2.1. Numerical characteristics of the random variable ............................................................................................. 10 LECTURE 3 ..................................................................................................................................................................... 15 3.1. Uniform distribution .......................................................................................................................................... 15 3.2. Exponential distribution ..................................................................................................................................... 16 3.3 Normal (Gauss) distribution ............................................................................................................................... 17 3.4 Сhi-square distribution ....................................................................................................................................... 20 3.5 Student distribution ............................................................................................................................................. 23 3.6 Fisher distribution............................................................................................................................................... 25 3.7 Gamma distribution ............................................................................................................................................ 26 LECTURE 4 ..................................................................................................................................................................... 28 4.1. The distribution law of the random argument function ...................................................................................... 28 4.2. Simulating of the random variable with the given distribution law by functional transformation ..................... 32 LECTURE 5 ..................................................................................................................................................................... 34 5.1. The law of large numbers ................................................................................................................................... 34 12.2. The central limit theorem ................................................................................................................................. 37 LECTURE 6 ..................................................................................................................................................................... 40 6.1. Mathematical statistics. Basic concepts ............................................................................................................. 40 6.2. The distribution law estimate ............................................................................................................................. 41 LECTURE 7 ..................................................................................................................................................................... 48 7.1. Point estimates of numerical characteristics ..................................................................................................... 48 LECTURE 8 ..................................................................................................................................................................... 52 8.1. The distribution parameters estimate ................................................................................................................. 52 8.2 Parameter estimation by the results of unequal measurements .......................................................................... 57 LECTURE 9 ..................................................................................................................................................................... 59 9.1 Interval estimate of numerical characteristics .................................................................................................... 59 9.2 The confidence interval for mathematical expectation ....................................................................................... 61 9.3 The confidence interval for dispersion ................................................................................................................ 62 9.4Theconfidence interval for probability ................................................................................................................. 63 LECTURE 10 ................................................................................................................................................................... 64 10.1 Statistical check of hypotheses. Classification of hypotheses ........................................................................... 64 10.2 Significance criterion ........................................................................................................................................ 65 10.3. Hypothesis testing about probabilities equality ............................................................................................... 69 LECTURE 11 ................................................................................................................................................................... 71 11.1. Hypothesis test about the distribution law. Fitting criteria ............................................................................. 71 11.2. Pirson criterion ................................................................................................................................................ 72 11.3. Kolmogorov criterion ....................................................................................................................................... 74 LECTURE 12 ................................................................................................................................................................... 78 12.1 Fitting criterion Mises-Smirnov criterion ......................................................................................................... 78 12.2 Neumann-Pirsona criterion .............................................................................................................................. 80 LECTURE 13 ................................................................................................................................................................... 85 13.1 Hypothesis test about the mathematical expectation at the known dispersion .................................................. 85 13.2 Hypothesis test about the mathematical expectation at the unknown dispersion .............................................. 88 13.3 Hypothesis test about the dispersion at the known mathematical expectation .................................................. 88 13.4 Hypothesis test about dispersionat the unknown mathematical expectation ..................................................... 90 13.5 Hypothesis test about mathematical expectation equality ................................................................................. 91 13.6 Hypothesis test about the dispersions equality.................................................................................................. 93 LECTURE 14 ................................................................................................................................................................... 95 14.1 Two-dimensional random variables. The two-dimensional distribution law .................................................... 95 14.2 Dependent and independent random variables ................................................................................................. 96 14.3 Conditional distribution laws ............................................................................................................................ 97 LECTURE 15 ................................................................................................................................................................... 98 15.1 Numerical characteristics of two-dimensional random variable ...................................................................... 98 15.2 Conditional numerical characteristics .............................................................................................................. 99 LECTURE 16 ................................................................................................................................................................
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