qualityTools (version 1.55)

weibull3: The Weibull Distribution (3 Parameter)

Description

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

Usage

dweibull3(x, shape, scale, threshold)
pweibull3(q, shape, scale, threshold)
qweibull3(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.

Value

dweibull3 gives the density, pweibull3 gives the distribution function and qweibull3 gives the quantile function.

Details

The Weibull 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)

References

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

See Also

uniroot

Examples

#Simple Example
dweibull3(x=1,scale=1,shape=5,threshold=0)
temp=pweibull3(q=1,scale=1,shape=5,threshold=0)
temp
qweibull3(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=dweibull3(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 Weibull-Distribution")
#lines(x,y=dweibull3(x,threshold=0,shape=1,scale=1),lwd=2,col="red")
#lines(x,y=dweibull3(x,threshold=0,shape=1.5,scale=2),lwd=2,col="blue")
#lines(x,y=dweibull3(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=pweibull3(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 Weibull-Distribution")
#lines(x,y=pweibull3(x,threshold=0,shape=1,scale=1),lwd=2,col="red")
#lines(x,y=pweibull3(x,threshold=0,shape=1.5,scale=2),lwd=2,col="blue")
#lines(x,y=pweibull3(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)