Enterprise Risk Management Formula Book
Appendix A.1: Probability Distributions:
Discrete (univariate) distributions
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A.1 Discrete (univariate) distributions:
Binomial (and Bernoulli), Poisson
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Distribution name
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Binomial
distribution
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Common notation
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Parameters
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 = number of
(independent) trials, positive integer
 = probability
of success in each trial, 
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Support
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Probability mass
function
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Cumulative distribution
function
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Mean
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Variance
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Skewness
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(Excess) kurtosis
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Characteristic function
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Other comments
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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  .
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Distribution name
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Poisson distribution
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Common notation
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Parameters
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 = event rate
( )
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Support
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Probability mass
function
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Cumulative distribution
function
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(can also be expressed using the incomplete gamma
function)
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Mean
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Variance
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Skewness
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(Excess) kurtosis
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Characteristic function
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Other comments
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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
 .
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