---

Contents

PART-I. PROBABILITY OF EVENTS
1 Introductory problems
2 Outcomes and events
3 Relative frequency and probability
4 Random numbers
5 Classical problems
6 Geometrical problems, uniform distributions
7 Basic properties of probability
8 Conditional relative frequency and conditional probability
9 Independence of events
10 *** Infinite sequences of events
11 *** Drawing with or without replacement. Permutations

PART-II. DISCRETE DISTRIBUTIONS
12 Discrete random variables and distributions
13 Uniform distribution (discrete)
14 Hyper-geometrical distribution
15 Binomial distribution
16 Geometrical distribution (pessimistic)
17 Geometrical distribution (optimistic)
18 *** Negative binomial distribution (pessimistic)
19 *** Negative binomial distribution (optimistic)
20 Poisson-distribution
21 Higher dimensional discrete random variables and distributions
22 *** Poly-hyper-geometrical distribution
23 *** Polynomial distribution
24 Generating a random variable with a given discrete distribution
25 Mode of a distribution
26 Expected value of discrete distributions
27 Expected values of the most important discrete distributions
28 Expected value of a function of a discrete random variable
29 Moments of a discrete random variable
30 Projections and conditional distributions for discrete distributions
31 Transformation of discrete distributions

PART-III. CONTINOUS DISTRIBUTIONS IN ONE-DIMENSION
32 Continuous random variables
33 Distribution function
34 *** Empirical distribution function
35 Density function
36 *** Histogram
37 Uniform distributions
38 Distributions of some functions of random numbers
39 *** Arc-sine distribution
40 *** Cauchy distribution
41 *** Beta distributions
42 Exponential distribution
43 *** Gamma distribution
44 Normal distributions
45 *** Distributions derived from normal
46 ***Generating a random variable with a given continuous distribution
47 Expected value of continuous distributions
48 Expected value of a function of a continuous random variable
49 ***Median
50 Standard deviation, etc.
51 *** Poisson-processes
52 ***Transformation from line to line

PART-IV. TWO-DIMENSIONAL CONTINOUS DISTRIBUTIONS
53 Two-dimensional random variables and distributions
54 Uniform distribution on a two-dimensional set
55 *** Beta distributions in two-dimensions
56 Projections and conditional distributions
57 Normal distributions in two-dimensions
58 Independence of random variables
59 Generating a two-dimensional random variable
60 Properties of the expected value, variance and standard deviation
61 Transformation from plane to line
62 *** Transformation from plane to plane
63 *** Sums of random variables. Convolution
64 Limit theorems to normal distributions

PART-V. STATISTICS
65 Regression in one-dimension
66 Regression in two-dimensions
67 Linear regression
68 Confidence intervals
69 U-tests
70 *** T-tests
71 *** Chi-square-test for fitness
72 *** Chi-test for standard deviation (Chi-square-test for variance)
73 *** F-test for equality of variances (of standard deviations)
74 *** Test with ANOVA (Analysis of variance)

PART-VI. LIST OF STATISTICAL EXCEL FUNCTIONS

PART-VII. ACKNOWLEDGEMENTS

---

Foreword

This is an introductory textbook to probability theory and statistics with the usual material taught at most universities. Its special feature, however, is that it contains interactive simulation files. These files are important, because the real life meaning of most of the notions of probability theory and statistics can be experienced only if we make a large number of experiments, not only once, but several times, and not only under a given set of conditions, but under modified conditions, as well.

The simulation files included in this textbook make it possible that the reader could see the results of many experiments, and could repeat them several times, and he or she could modify the parameters of the problem, as well. Since the simulation files are written in Excel, students themselves can easily construct similar simulation files. Their activity will increase their confidence and interest in the subject.

The book consists of five parts:
1. Probability of events
2. Discrete distributions
3. Continuous distributions in one-dimension
4. Two-dimensional continuous distributions
5. Statistics

The author is devoted to write an exercise-book soon, which will - hopefully - help the students to learn not only the probabilistic and statistic notions but the necessary Excel tricks to construct simulation files according to their own needs.

---