Statistics 241

### Statistics 241/541: Probability Theory

YALE UNIVERSITY, FALL 1995
Prof. Nicolas Hengartner
DESCRIPTION
This course serves for an introduction to probability theory, for students who have had calculus (but not advanced calculus or measure theory). It serves as prerequisite for Statistics 242. The course emphasizes the aspects of probability theory that are needed to use probability models in applications. The material is presented from the concrete to the abstract. Students should take away from this course the following:
• Probability spaces as models for chance experiments; the probability of an event as a quantity representing its long-run relative frequency
• The notion of random variable: a function on an outcome set, used to model a quantity determined by the outcome of a chance experiment.
• The expected value of a random variable, representing its expected long-run average value .
• The limit theorems: weak law of large numbers, strong law of large numbers and the central limit theorem.
• The idea of a distribution as an assignment of probability to intervals, and thence to sets that can be made from intervals by set operations; the distribution of a random variable.
• The two main kinds of distributions: discrete and absolutely continuous, and the various kinds of calculations involving them.
• The standard "brand-name" distributions and models most used in practice, including the Poisson process.
• Joint, marginal and conditional distributions.
• Distribution theory: finding the distribution of a function of one or more random variables; including the use of probability generating functions.
• Conditional probability and independence; the use of conditioning and Bayes's Theorem.
TEXTBOOK:
Introduction to Probabily, D.G. Kelly, MacMillan press, 1994.