# dgamma3: The gamma Distribution (3 Parameter) In qualityTools: Statistical Methods for Quality Science

## Description

Density function, distribution function and quantile function for the gamma distribution.

## Usage

 ```1 2 3``` ```dgamma3(x, shape, scale, threshold) pgamma3(q, shape, scale, threshold) qgamma3(p, shape, scale, threshold, ...) ```

## Arguments

 `x, q` vector of quantiles `p` vector of probabilities `shape` shape parameter by default 1 `scale` scale parameter by default 1 `threshold` threshold parameter by default 0 `...` Arguments that can be passed into `uniroot`.

## Details

The gamma distribution with ‘scale’ parameter alpha, ‘shape’ parameter c and ‘threshold’ parameter zeta has density given by

f(x) = (c/alpha) (((x-zeta)/alpha)^(c-1)) exp(-((x-zeta)/alpha)^c)

The cumulative distribution function is given by

F(x) = 1 - exp(-((x-zeta)/alpha)^c)

## Value

`dgamma3` gives the density, `pgamma3` gives the distribution function and `qgamma3` gives the quantile function.

## Note

`qgamma3` calls `uniroot` for each value of the argument ‘p’. The solution is consequently not exact; the ... can be used to obtain a more accurate solution if necessary.

## Author(s)

Thomas Roth thomas.roth@tu-berlin.de
Etienne Stockhausen stocdarf@mailbox.tu-berlin.de

## References

Johnson, L., Kotz, S., Balakrishnan, N. (1995) Continuous Univariate Distributions-Volume 1, 2nd ed. New York: John Wiley & Sons.

`uniroot`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51``` ```##Simple Example dgamma3(x=1,scale=1,shape=5,threshold=0) temp=pgamma3(q=1,scale=1,shape=5,threshold=0) temp qgamma3(p=temp,scale=1,shape=5,threshold=0) # ##Visualized Example ##prepare screen #dev.new() #split.screen(matrix(c(0,0.5,0,1, 0.5,1,0,1),byrow=TRUE,ncol=4)) ##generate values #x=seq(0,3,length=1000) ##plot different density functions #screen(1) #plot(x,y=dgamma3(x,threshold=0,shape=0.5,scale=1),col="green", # xlim=c(0,2.5),ylim=c(0,2.5),type="l",lwd=2,xlab="x", # ylab="f(x)",main="Density Function of gamma-Distribution") #lines(x,y=dgamma3(x,threshold=0,shape=1,scale=1),lwd=2,col="red") #lines(x,y=dgamma3(x,threshold=0,shape=1.5,scale=2),lwd=2,col="blue") #lines(x,y=dgamma3(x,threshold=0,shape=5,scale=1),lwd=2,col="orange") ##add legend #legend("topright",legend=c(expression(paste(alpha, " = 1 ")* # paste(c, " = 0.5 ")*paste(zeta," = 0")), # expression(paste(alpha, " = 1 ")*paste(c, " = 1 ")* # paste(zeta," = 0")),expression(paste(alpha, " = 2 ")* # paste(c, " = 1.5 ")*paste(zeta," = 0")), # expression(paste(alpha, " = 1 ")*paste(c, " = 5 ")* # paste(zeta," = 0"))),col=c("green","red","blue","orange"), # text.col="black",lwd=2,bty="0",inset=0.04) #abline(v=0,lty=2,col="grey") #abline(h=0,lty=2,col="grey") ##plot different distribution functions #screen(2) #plot(x,y=pgamma3(x,threshold=0,shape=0.5,scale=1),col="green", # xlim=c(0,2.5),ylim=c(0,1),type="l",lwd=2,xlab="x",ylab="F(x)", # main="Cumulative Distribution Function of gamma-Distribution") #lines(x,y=pgamma3(x,threshold=0,shape=1,scale=1),lwd=2,col="red") #lines(x,y=pgamma3(x,threshold=0,shape=1.5,scale=2),lwd=2,col="blue") #lines(x,y=pgamma3(x,threshold=0,shape=5,scale=1),lwd=2,col="orange") ##add legend #legend("bottomright",legend=c(expression(paste(alpha, " = 1 ")* # paste(c, " = 0.5 ")*paste(zeta," = 0")), # expression(paste(alpha, " = 1 ")*paste(c, " = 1 ")* # paste(zeta," = 0")),expression(paste(alpha, " = 2 ")* # paste(c, " = 1.5 ")*paste(zeta," = 0")), # expression(paste(alpha, " = 1 ")*paste(c, " = 5 ")* # paste(zeta," = 0"))),col=c("green","red","blue","orange"), # text.col="black",lwd=2,bty="0",inset=0.04) #abline(v=0,lty=2,col="grey") #abline(h=0,lty=2,col="grey") #close.screen(all=TRUE) ```

### Example output

```Loading required package: Rsolnp

Attaching package: ‘qualityTools’

The following object is masked from ‘package:stats’:

sigma

[1] 0.01532831
[1] 0.003659847
[1] 0.9999981
```

qualityTools documentation built on May 2, 2019, 10:21 a.m.