## TAU:0365-2100 | ## 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.

- 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.

- Axiomatic approach:
- 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).

- A monotone function on (0,1) as a random variable.
- TRANSFORMATIONS.

Full text: Postscript.- One-dimensional transformations: linear and nonlinear; smooth and non-smooth; monotone and non-monotone.

- 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.

- 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.

- 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.

- One- and two-dimensional distributions in general;
- CONDITIONING.

Short summary: Postscript;

full text: Postscript.- Conditional distribution.
- Total probability formulas and Bayes formulas for various cases (discrete, absolutely continuous, singular).

- SPECIAL DISTRIBUTIONS.

Full text: Postscript.- One-dimensional: Gamma, Chi-square, Student, Fisher. Their connections to exponential, normal, Poisson distributions.
- Two-dimensional: normal correlation.

- CONVERGENCE.
- Convergence of a sequence of random variables.
- Convergence of expectations. Monotone convergence theorem. Dominated convergence theorem.

- LIMIT THEOREMS.
- Borel-Cantelli lemma(s).
- Strong law of large numbers.
- Central limit theorem.