Columbia University in the City of New York              |
 
DEPARTMENT OF STATISTICS 
 
 
New York, N.Y. 10027
 
618 Mathematics Building 
TEL: (212) 854-3652/3 
FAX: (212) 663-2454 

W 4105: PROBABILITY

Spring Semester 1999

Professor Ioannis Karatzas

 

COURSE SYLLABUS

 

THE CONCEPT OF PROBABILITY. Historical sketch. Classical, Frequentist, and Axiomatic definitions.

UNIFORM PROBABILITY MODELS. Principles of Combinatorial Analysis.

CONDITIONAL PROBABILITY AND INDEPENDENCE. The Bayes rule.

SEQUENCES OF INDEPENDENT TRIALS. The Binomial and Multinomial distributions. Limit Theorems of deMoivre-Laplace and Poisson. The Poisson distribution. The normal distribution.

RANDOM VARIABLES AND DISTRIBUTION FUNCTIONS. Definitions. Properties. Examples of discrete and continuous distributions. Multivariate distributions. Independence of random variables.

TRANSFORMATIONS OF RANDOM VARIABLES.

NUMERICAL CHARACTERISTICS OF RANDOM VARIABLES. Expectation. Median. Variance and covariance. The Markov and Chebyshev inequalities.

LIMIT THEOREMS OF PROBABILITY. The Law of Large Numbers. Moment-generating functions. The Central Limit Theorem. Applications. Cramer's Theorem.

CONDITIONAL DISTRIBUTIONS. The Law of Total Probability. Applications: Simple random Walk, the gambler's ruin problem.

 

Required Text:

        S. ROSS      "A First Course in Probability". 5th Edition, Prentice-Hall (1998).


Recommended Texts:

        W. FELLER      "An Introduction to Probability Theory and Its Applications".   Third Edition, J. Wiley & Sons (1967).
        B. GNEDENKO      "The Theory of Probability".   4th Printing, Mir Publishers, Moscow (1978).

 

Class Schedule: