Manuals/calci/GAMMADIST

GAMMADIST(x,alpha,beta,cumulative,accuracy)

• $x$ is the value of the distribution.
• $alpha$ and $beta$ are the value of the parameters.
• $cumulative$ is the logical value like true or false.
• $accuracy$ gives accurate value of the solution.

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 $GAMMADIST(x,alpha,beta,cumulative,accuracy)$, $x$ is the value of the distribution, $\alpha$ is called shape parameter and $\beta$ is the rate parameter of the distribution and $cumulative$ is the logical value like TRUE or FALSE.
• If $cumulative$ is TRUE, then this function gives the Cumulative Distribution value and if is FALSE then it gives the Probability Density Function.
• $cumulative$ gives accurate value of the solution.
• 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.

• The Standard Gamma Probability Density function is:

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

• The Cumulative Distribution Function of Gamma is :

$F(x;\alpha ,\beta )={\frac {\gamma (\alpha ,{\frac {x}{\beta }})}{\Gamma (\alpha )}}$, 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 $\beta \leq 0$.

ZOS Section

• The syntax is to calculate GAMMADIST in ZOS is $GAMMADIST(x,alpha,beta,cumulative,accuracy)$.
  • $x$ is the value of the distribution,
  • $alpha$ and $beta$ are the value of the parameters
  • $cumulative$ is the logical value like true or false.
  • $accuracy$ gives accurate value of the solution.
• For e.g.,

Examples

1. GAMMADIST(8.15372,5,7,TRUE)=0.006867292
2. GAMMADIST(20.78542,2,6,TRUE)=0.860283293
3. GAMMADIST(20.78542,2,6,FALSE)=0.01806997
4. GAMMADIST(45.6523,9,4,FALSE)=0.019724471
5. GAMMADIST(8.15372,5,7,TRUE,0.5)= 0.00693316259
6. GAMMADIST(8.15372,5,7,TRUE,0.9)=0.0067648564

See Also

• DATE
• DAYS360
• DATEVALUE

References

Gamma Distribution