Encyclopedia of Financial Models III

Discrete Probability Distributions

MARKUS HÖCHSTÖTTER, PhD

Assistant Professor, University of Karlsruhe

SVETLOZAR T. RACHEV, PhD, Dr Sci

Frey Family Foundation Chair-Professor, Department of Applied Mathematics and Statistics, Stony Brook University, and Chief Scientist, FinAnalytica

FRANK J. FABOZZI, PhD, CFA, CPA

Professor of Finance, EDHEC Business School

Abstract:Discrete probability distributions are needed whenever the random variable is to describe a quantity that can assume values from a countable set, either finite or infinite. A discrete probability distribution (or law) is quite intuitive in that it assigns certain values positive probabilities adding up to one, while any other value automatically has zero probability. In general, neglecting some of the mathematical rigor, discrete distributions can be understood from the insight gained from descriptive statistics. For example, the random number of defaults in a bond portfolio inside of a given period of time can be modeled with a discrete probability distribution. Another example is given by sampling when we are interested in whether an observation belongs to a certain group. Also, simple stock price models are based on discrete laws where the stock price can only change to one of a finite number of possible values.

Discrete random variables are random variables on the countable space. We present the most important discrete random variables used in finance and their probability distribution (also called probability ...