Discrete Probability Distributions

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. Keywords: Discrete random variables; probability distribution; probability law; discrete law; discrete cumulative distribution; variance; standard deviation; Bernoulli distribution; drawing with replacement; binomial distribution; binomial coefficient; binomial tree; hypergeometric; multinomial distribution; discrete uniform distribution; Poi; binomial coefficient; factorial; multinomial coefficient; polynomial coefficient; path-dependent; Markov