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
|
Common notation
|
![](I/ERMFormulaBookAppendixDiscrete_files/image001.png)
|
Parameters
|
= number of
(independent) trials, positive integer
= probability
of success in each trial, ![](I/ERMFormulaBookAppendixDiscrete_files/image004.png)
|
Support
|
![](I/ERMFormulaBookAppendixDiscrete_files/image005.png)
|
Probability mass
function
|
![](I/ERMFormulaBookAppendixDiscrete_files/image006.png)
|
Cumulative distribution
function
|
![](I/ERMFormulaBookAppendixDiscrete_files/image007.png)
|
Mean
|
![](I/ERMFormulaBookAppendixDiscrete_files/image008.png)
|
Variance
|
![](I/ERMFormulaBookAppendixDiscrete_files/image009.png)
|
Skewness
|
![](I/ERMFormulaBookAppendixDiscrete_files/image010.png)
|
(Excess) kurtosis
|
![](I/ERMFormulaBookAppendixDiscrete_files/image011.png)
|
Characteristic function
|
![](I/ERMFormulaBookAppendixDiscrete_files/image012.png)
|
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:
![](I/ERMFormulaBookAppendixDiscrete_files/image014.png)
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
|
![](I/ERMFormulaBookAppendixDiscrete_files/image024.png)
|
Parameters
|
= event rate
( )
|
Support
|
![](I/ERMFormulaBookAppendixDiscrete_files/image027.png)
|
Probability mass
function
|
![](I/ERMFormulaBookAppendixDiscrete_files/image028.png)
|
Cumulative distribution
function
|
![](I/ERMFormulaBookAppendixDiscrete_files/image029.png)
(can also be expressed using the incomplete gamma
function)
|
Mean
|
![](I/ERMFormulaBookAppendixDiscrete_files/image025.png)
|
Variance
|
![](I/ERMFormulaBookAppendixDiscrete_files/image025.png)
|
Skewness
|
![](I/ERMFormulaBookAppendixDiscrete_files/image030.png)
|
(Excess) kurtosis
|
![](I/ERMFormulaBookAppendixDiscrete_files/image031.png)
|
Characteristic function
|
![](I/ERMFormulaBookAppendixDiscrete_files/image032.png)
|
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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