Quantitative Risk Management: A Practical Guide to Financial Risk, + Website

Appendix 11.1: Probability Distributions

Binomial

The binomial distribution counts the number of successes in a sequence of independent yes/no or succeed/fail (Bernoulli) trials. With p = probability of success, q = 1 − p = probability of failure, the probability of k successes out of n trials is:

where

For q = 0.01, n = 100, P[k = 0] = 0.366, P[k = 1] = 0.370, P[k = 2] = 0.185, P[k ≥ 3] = 0.079

Poisson

The Poisson distribution gives the probability of observing j events during a fixed time period, when events occur at a fixed rate per unit of time and independently over time. If the intensity (or average rate per unit of time) is λ, then the probability that j events occur is:

Gamma

A gamma random variable is a positive random variable with density

Negative Binomial

The negative binomial is a discrete distribution (like the binomial taking values 0, 1, 2,...). The initial definition arises, like the binomial, when considering Bernoulli trials each of which may be either a success ...