Enterprise Risk Management Formula Book
1. Function Definitions
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1.1 Gamma function, ![](I/ERMFormulaBookFunctionDefinitions_files/image001.png)
![](I/ERMFormulaBookFunctionDefinitions_files/image002.png)
Defined for
,
not
a negative integer. Other properties:
. If
is
a positive integer then
.
.
1.2 Incomplete gamma
function,
, beta function,
, incomplete beta function,
, regularised
incomplete beta function, ![](I/ERMFormulaBookFunctionDefinitions_files/image012.png)
![](I/ERMFormulaBookFunctionDefinitions_files/image013.png)
![](I/ERMFormulaBookFunctionDefinitions_files/image014.png)
![](I/ERMFormulaBookFunctionDefinitions_files/image015.png)
![](I/ERMFormulaBookFunctionDefinitions_files/image016.png)
The beta function is related to the gamma function as
follows:
![](I/ERMFormulaBookFunctionDefinitions_files/image017.png)
The gamma, incomplete gamma, beta, incomplete beta and
regularised incomplete beta can also be defined for negative (non-integral)
values of
and
and
for complex (non-real) values by analytic continuation.
1.3 The binomial coefficient,
![](I/ERMFormulaBookFunctionDefinitions_files/image019.png)
For
and
integers
this is
defined as:
![](I/ERMFormulaBookFunctionDefinitions_files/image022.png)
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