### Probability Theory - old version

This is an old (1999) version of the second-year course probability theory. It is more formal in style than the recent version. It is full of proofs. If this is what you like, take it.

#### TOPICS

1. PROBABILITY SPACE.
Full text: Postscript.
• Axiomatic approach:
definition of a sigma-field and a probability measure.
• Analytic approach:
the interval (0,1) as a continuous probability space;
smooth functions as random variables;
cumulative distribution function and density.
2. DISTRIBUTION FUNCTION AND QUANTILE FUNCTION.
Short summary: Postscript;
full text: Postscript.
• A monotone function on (0,1) as a random variable.
Cumulative distribution function and quantile function.
Atoms and gaps.
• Convergence in distribution.
Special distributions:
discrete (uniform, binomial, Poisson, geometric) revisited,
and continuous (uniform, normal, exponential).
3. TRANSFORMATIONS.
Full text: Postscript.
• One-dimensional transformations: linear and nonlinear; smooth and non-smooth; monotone and non-monotone.
4. MATHEMATICAL EXPECTATION AND INTEGRAL.
Short summary: Postscript;
full text: Postscript.
• Beyond Riemann integrability.
• Discrete approximation of a continuous random variable.
• Expectation: definition and properties.
• Expectation and density.
• Expectation and quantile function.
• Expectation and cumulative distribution function.
• Unbounded case, truncation, integrability.
• Expectation of a function of a random variable.
• Variance, moments, moment generating function.
• Examples: special distributions.
5. BOREL SETS, FUNCTIONS, AND MEASURES.
Short summary: Postscript;
full text: Postscript.
• Intervals and elementary sets.
• More complicated sets.
• Borel sets.
• Borel functions.
• Borel measures, Lebesgue measure.
• Lebesgue integral.
6. DISTRIBUTIONS.
Short summary: Postscript;
full text: Postscript.
• One- and two-dimensional distributions in general;
support, atoms, discrete part and continuous part.
• One- and two-dimensional density;
absolutely continuous part and singular part.
• Marginal distribution.
• Independence.
• Regression and correlation.
• Transformations.
• Distribution of sum, product, quotient.
7. CONDITIONING.
Short summary: Postscript;
full text: Postscript.
• Conditional distribution.
• Total probability formulas and Bayes formulas for various cases (discrete, absolutely continuous, singular).
8. SPECIAL DISTRIBUTIONS.
Full text: Postscript.
• One-dimensional: Gamma, Chi-square, Student, Fisher. Their connections to exponential, normal, Poisson distributions.
• Two-dimensional: normal correlation.
9. CONVERGENCE.
• Convergence of a sequence of random variables.
• Convergence of expectations. Monotone convergence theorem. Dominated convergence theorem.
10. LIMIT THEOREMS.
• Borel-Cantelli lemma(s).
• Strong law of large numbers.
• Central limit theorem.