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.