# dlnorm3: The Lognormal Distribution (3 Parameter) In qualityTools: Statistical Methods for Quality Science

## Description

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

## Usage

 ```1 2 3``` ```dlnorm3(x, meanlog, sdlog, threshold) plnorm3(q, meanlog, sdlog, threshold) qlnorm3(p, meanlog, sdlog, threshold, ...) ```

## Arguments

 `x, q` vector of quantiles `p` vector of probabilities `meanlog, sdlog` mean and standard deviation of the distribution on the log scale with default values of ‘0’ and ‘1’ respectively. `threshold` threshold parameter by default 0 `...` Arguments that can be passed into `uniroot`.

## Details

The Lognorm distribution with ‘meanlog’ parameter zeta, ‘meansd’ parameter sigma and ‘threshold’ parameter theta has density given by

f(x) = (1/(sqrt(2*pi)*sigma*(x-theta)))
*exp(-(((log((x-theta))-zeta)^2)/(2*(sigma)^2)))

The cumulative distribution function is given by

F(x) = pnorm((log((x-theta))-zeta)/sigma)

## Value

`dlnorm3` gives the density, `plnorm3` gives the distribution function and `qlnorm3` gives the quantile function.

## Note

`qlnorm3` 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 [email protected]
Etienne Stockhausen [email protected]

## References

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

`uniroot`
 ``` 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``` ```#Simple Example dlnorm3(x=2,meanlog=0,sdlog=1/8,threshold=1) temp=plnorm3(q=2,meanlog=0,sdlog=1/8,threshold=1) temp qlnorm3(p=temp,meanlog=0,sdlog=1/8,threshold=1) # ##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(0,4,length=1000) ##plot different density functions #screen(1) #plot(x,y=dlnorm3(x,0,1/8,1),col="green",xlim=c(0,3),type="l",lwd=2,xlab="x", # ylab="f(x)",main="Density Function of Log-Normal-Distribution") #lines(x,y=dlnorm3(x,0,0.5,0),lwd=2,col="red") #lines(x,y=dlnorm3(x,0,1,0),lwd=2,col="blue") #lines(x,y=dlnorm3(x,1,1/8,0),lwd=2,col="orange") ##add legend #legend("topleft",legend=c(expression(paste(zeta," = 0 ")* # paste(sigma, " = 1/8 ")*paste(theta," = 1")), # expression(paste(zeta," = 0 ")*paste(sigma, " = 0.5 ")* # paste(theta," = 0")),expression(paste(zeta," = 0 ")* # paste(sigma, " = 1 ")*paste(theta," = 0")), # expression(paste(zeta," = 1 ")*paste(sigma, " = 1/8 ")* # paste(theta," = 0"))),col=c("green","red","blue","orange"), # text.col="black",lwd=2,bty="o",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=plnorm3(x,0,1/8,1),col="green",xlim=c(0,3),type="l",lwd=2,xlab="x", # ylab="F(x)", # main="Cumulative Distribution Function of Log-Normal-Distribution") #lines(x,y=plnorm3(x,0,0.5,0),lwd=2,col="red") #lines(x,y=plnorm3(x,0,1,0),lwd=2,col="blue") #lines(x,y=plnorm3(x,1,1/8,0),lwd=2,col="orange") ##add legend #legend("topleft",legend=c(expression(paste(zeta," = 0 ")* # paste(sigma, " = 1/8 ")*paste(theta," = 1")), # expression(paste(zeta," = 0 ")*paste(sigma, " = 0.5 ")* # paste(theta," = 0")),expression(paste(zeta," = 0 ")* # paste(sigma, " = 1 ")*paste(theta," = 0")), # expression(paste(zeta," = 1 ")*paste(sigma, " = 1/8 ")* # paste(theta," = 0"))),col=c("green","red","blue","orange"), # text.col="black",lwd=2,bty="o",inset=0.04) #abline(v=0,lty=2,col="grey") #abline(h=0,lty=2,col="grey") #close.screen(all=TRUE) ```