Manuals/calci/GAMMADIST

GAMMADIST(x,alpha,beta,cu)

Description

This function gives the value of the Gamma Distribution.
The Gamma Distribution can be used in a queuing models like, the amount of rainfall accumulated in a reservoir.
This distribution is the Continuous Probability Distribution with two parameters $\alpha$ and $\beta$ .
In $GAMMADIST(x,alpha,beta,cu)$ , $x$ is the value of the distribution, $\alpha$ is called shape parameter and $\beta$ is the rate parameter of the distribution and $cu$ is the logical value like TRUE or FALSE.
If $cu$ is TRUE, then this function gives the Cumulative Distribution value and if is FALSE then it gives the Probability Density Function.
The gamma function is defined by :

$Gamma(t)=\int \limits _{0}^{\infty }x^{t-1}e^{-x}dx$ .

It is for all complex numbers except the negative integers and zero.
The Probability Density Function of Gamma function using Shape, rate parameters is:

$f(x;\alpha ,\beta )={\frac {x^{\alpha -1}e^{-{\frac {x}{\beta }}}}{\beta ^{\alpha }\Gamma (\alpha )}}$ , for

x,\alpha ,\beta >0

, where

e

is the natural number(e = 2.71828...),

\alpha

is the number of occurrences of an event, and

\Gamma (\alpha )

is the Gamma function.

$f(x,\alpha )={\frac {x^{\alpha -1}e^{-x}}{\Gamma (\alpha )}}$ .

$F(x;\alpha ,\beta )={\frac {\gamma (\alpha ,{\frac {x}{\beta }})}{\Gamma (\alpha )}}$ , or

F(x;\alpha ,\beta )=e^{-{\frac {x}{\beta }}}\sum _{i=k}^{\infty }{\frac {1}{i!}}({\frac {x}{\beta }})^{i}

for any positive integer

k

.

When alpha is a positive integer, then the distribution is called Erlang distribution.
If the shape parameter $\alpha$ is held fixed, the resulting one-parameter family of distributions is a natural exponential family.
For a positive integer $n$ , when $\alpha ={\frac {n}{2}}$ , $\beta =2$ , and $cu=TRUE$ , GAMMADIST returns (1 - CHIDIST(x)) with $n$ degrees of freedom.
This function shows the result as error when

1.Any one of the argument is non numeric
2. $x<0$ ,  $\alpha \leq 0$  or  $\beta \leq 0$