Mathematical tools

1. Set theory ( Exercise set 1)

2. Combinatorics

   1. Permutations ( Exercise set 1)

   2. k-permutations ( Exercise set 1)

   3. Combinations ( Exercise set 1)

   4. Partitions ( Exercise set 1)

3. Sequences

4. Limit of a sequence

5. Review of differentiation rules ( Test 1; Test 2; Test 3; Test 4)

6. Review of integration rules ( Exercise set 1)

7. Computing the Riemann-Stieltjes integral: some rules ( Exercise 1)

8. Special functions

   1. The Gamma function ( Exercise set 1)

   2. The Beta function ( Exercise set 1)

Fundamentals of probability

Probability and events

1. Probability and probability measures ( Exercise set 1)

2. Zero-probability events

3. Conditional probability ( Exercise set 1)

4. Bayes' rule ( Exercise set 1)

5. Independent events ( Exercise set 1)

6. Multiple choice tests: Test 1

Random variables and random vectors

1. Random variables and univariate probability distributions ( Exercise set 1; Exercise set 2)

2. Random vectors and multivariate probability distributions ( Exercise set 1; Exercise set 2)

3. Expected value ( Exercise set 1; Exercise set 2)

4. Expected value and the Lebesgue integral

5. Properties of the expected value ( Exercise set 1)

6. Variance ( Exercise set 1; Exercise set 2)

7. Covariance ( Exercise set 1; Exercise set 2)

8. Linear correlation coefficient ( Exercise set 1)

9. Covariance matrix ( Exercise set 1)

10. Indicator functions ( Exercise set 1)

11. Multiple choice tests: Test 2, Test 3

Conditional probabilities, conditional distributions and independence

1. Conditional probability as a random variable

2. Conditional probability distributions ( Exercise set 1)

3. Conditional expectation

4. Independent random variables ( Exercise set 1)

Additional topics in probability theory

1. Basic probabilistic inequalities ( Exercise set 1)

2. Legitimate probability mass functions ( Exercise set 1)

3. Legitimate probability density functions ( Exercise set 1)

4. Factorization of joint probability mass functions

5. Factorization of joint probability density functions

6. Functions of random variables and their distribution ( Exercise set 1)

7. Functions of random vectors and their distribution

8. Moments of a random variable

9. Cross-moments of a random vector

10. Moment generating function of a random variable ( Exercise set 1)

11. Moment generating function of a random vector ( Exercise set 1)

12. Characteristic function of a random variable ( Exercise set 1)

13. Characteristic function of a random vector

14. Sums of independent random variables

Probability distributions

Univariate discrete distributions

1. Bernoulli distribution ( Exercise set 1)

2. Binomial distribution ( Exercise set 1)

3. Poisson distribution ( Exercise set 1)

Univariate continuous distributions

1. Uniform distribution ( Exercise set 1; probability plots)

2. Exponential distribution ( Exercise set 1; probability plots)

3. Normal distribution ( Exercise set 1)

   1. Probability plots

   2. Values and tables

4. Chi-square distribution ( Exercise set 1)

   1. Probability plots

   2. Values and tables

5. Gamma distribution ( Exercise set 1; probability plots)

6. Student's t-distribution ( Exercise set 1; probability plots)

7. F distribution ( probability plots)

Multivariate discrete distributions

1. Multinomial distribution

Multivariate continuous distributions

1. Multivariate normal distribution

   1. Linear combinations

   2. Partitioned vectors

   3. Quadratic forms

2. Multivariate Student's t distribution

3. Wishart distribution

Multiple choice tests on probability distributions

1. Test 1, Test 2

Asymptotic theory

1. Sequences of random variables

2. Limit of a sequence of random variables

3. Sequences of random vectors and their convergence

4. Pointwise convergence ( Exercise set 1)

5. Almost sure convergence

6. Convergence in probability ( Exercise set 1)

7. Mean-square convergence

8. Convergence in distribution

9. Relations between modes of convergence

10. Laws of Large Numbers

11. Central Limit Theorems ( Exercise set 1)

12. Convergence of transformations ( Exercise set 1)

Fundamentals of statistics

1. Statistical inference

2. Point estimation

   1. Point estimation of the mean

   2. Point estimation of the variance

3. Set estimation

   1. Set estimation of the mean ( Exercise set 1)

   2. Set estimation of the variance ( Exercise set 1)

4. Hypothesis testing

   1. Testing hypotheses about the mean

   2. Testing hypotheses about the variance

5. Estimation methods

   1. Maximum likelihood

Glossary of probability and statistics terms

Miscellanea - Not on the front page

Probability exercises

Probability plots

Links to other e-books, digital textbooks and educational websites

Probability books / Statistics books

Random thoughts by StatLect's webmaster

Survey on paper textbook

Time varying regressions