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
Distribution name
|
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 .
|
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
|

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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
.
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