Manuals/calci/GAMMADIST

GAMMADIST(x,alpha,beta,cu)

• $x$ is the value of the distribution,
• $alpha$ and $beta$ are the value of the parameters
• $cu$ is the logical value like true or false.

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 :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x; \alpha,\beta)=\frac{x^{\alpha-1} e^{-\frac {x}{\beta}}}{\beta^{\alpha} Gamma(\alpha)}} , for

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, \alpha , \beta > 0 } , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} is the natural number(e = 2.71828...), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is the number of occurrences of an event, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Gamma(\alpha)} is the Gamma function.

• The standard Gamma Probability Density function is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,\alpha)=\frac{x^{\alpha-1} e^{-x}}{Gamma(\alpha)}} .

• The Cumulative Distribution Function of Gamma is :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x;\alpha,\beta)=\frac{\gamma(\alpha,\frac{x}{\beta})}{Gamma(\alpha)}} , or

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x;\alpha,\beta)= e^{-\frac {x}{\beta}} \sum_{i=k}^{\infty} \frac{1}{i!} (\frac{x}{\beta})^i} for any positive integer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} .

• When alpha is a positive integer, then the distribution is called Erlang distribution.
• If the shape parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is held fixed, the resulting one-parameter family of distributions is a natural exponential family.
• For a positive integer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha =\frac{n}{2}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta = 2} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cu= TRUE} , GAMMADIST returns (1 - CHIDIST(x)) with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} degrees of freedom.
• This function shows the result as error when

1.Any one of the argument is non numeric
2.Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x<0}
, $\alpha \leq 0$ or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta \le 0}

See Also

• DATE
• DAYS360
• DATEVALUE

References

Bessel Function