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### Enterprise Risk Management Formula Book Appendix A.1: Probability Distributions: Discrete (univariate) distributions

A.1         Discrete (univariate) distributions:

Binomial (and Bernoulli), Poisson

 Distribution name Binomial distribution Common notation Parameters = number of (independent) trials, positive integer  = probability of success in each trial, Support Probability mass function Cumulative distribution function Mean Variance Skewness (Excess) kurtosis Characteristic function Other comments Corresponds to the number of successes in a sequence of  independent experiments each of which has a probability  of being successful.   The Bernoulli distribution is  and corresponds to the likelihood of success of a single experiment.  Its probability mass function and cumulative distribution function are:   The Bernoulli distribution with , i.e. , has the minimum possible excess kurtosis, i.e. .   The mode of   is  if  is 0 or not an integer and is  if . If  then the distribution is bi-modal, with modes  and .

 Distribution name Poisson distribution Common notation Parameters = event rate () Support Probability mass function Cumulative distribution function (can also be expressed using the incomplete gamma function) Mean Variance Skewness (Excess) kurtosis Characteristic function Other comments Expresses the probability of a given number of events occurring in a fixed interval of time if the events occur with a known average rate and independently of the time since the last event.   The median is approximately .   The mode is  if  is not integral. Otherwise the distribution is bi-modal with modes  and .