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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 .