fitODBOD (version 1.5.3)

dTRI: Triangular Distribution Bounded Between [0,1]

Description

These functions provide the ability for generating probability density values, cumulative probability density values and moments about zero values for the Triangular Distribution bounded between [0,1].

Usage

dTRI(p,mode)

Value

The output of dTRI gives a list format consisting

pdf probability density values in vector form.

mean mean of the unit bounded Triangular distribution.

variance variance of the unit bounded Triangular distribution

Arguments

p

vector of probabilities.

mode

single value for mode.

Details

Setting \(min=0\) and \(max=1\) \(mode=c\) in the Triangular distribution a unit bounded Triangular distribution can be obtained. The probability density function and cumulative density function of a unit bounded Triangular distribution with random variable P are given by

$$g_{P}(p)= \frac{2p}{c} $$ ; \(0 \le p < c\) $$g_{P}(p)= \frac{2(1-p)}{(1-c)} $$ ; \(c \le p \le 1\) $$G_{P}(p)= \frac{p^2}{c} $$ ; \(0 \le p < c\) $$G_{P}(p)= 1-\frac{(1-p)^2}{(1-c)} $$ ; \(c \le p \le 1\) $$0 \le mode=c \le 1$$

The mean and the variance are denoted by $$E[P]= \frac{(a+b+c)}{3}= \frac{(1+c)}{3} $$ $$var[P]= \frac{a^2+b^2+c^2-ab-ac-bc}{18}= \frac{(1+c^2-c)}{18} $$

Moments about zero is denoted as $$E[P^r]= \frac{2c^{r+2}}{c(r+2)}+\frac{2(1-c^{r+1})}{(1-c)(r+1)}+\frac{2(c^{r+2}-1)}{(1-c)(r+2)} $$ \(r = 1,2,3,...\)

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.

References

horsnell1957economicalfitODBOD johnson1995continuousfitODBOD karlis2008polygonalfitODBOD okagbue2014usingfitODBOD

Examples

#plotting the random variables and probability values
col <- rainbow(4)
x <- seq(0.2,0.8,by=0.2)
plot(0,0,main="Probability density graph",xlab="Random variable",
ylab="Probability density values",xlim = c(0,1),ylim = c(0,3))
for (i in 1:4)
{
lines(seq(0,1,by=0.01),dTRI(seq(0,1,by=0.01),x[i])$pdf,col = col[i])
}

dTRI(seq(0,1,by=0.05),0.3)$pdf     #extracting the pdf values
dTRI(seq(0,1,by=0.01),0.3)$mean    #extracting the mean
dTRI(seq(0,1,by=0.01),0.3)$var     #extracting the variance

#plotting the random variables and cumulative probability values
col <- rainbow(4)
x <- seq(0.2,0.8,by=0.2)
plot(0,0,main="Cumulative density graph",xlab="Random variable",
ylab="Cumulative density values",xlim = c(0,1),ylim = c(0,1))
for (i in 1:4)
{
lines(seq(0,1,by=0.01),pTRI(seq(0,1,by=0.01),x[i]),col = col[i])
}

pTRI(seq(0,1,by=0.05),0.3)      #acquiring the cumulative probability values
mazTRI(1.4,.3)                  #acquiring the moment about zero values
mazTRI(2,.3)-mazTRI(1,.3)^2     #variance for when is mode 0.3

#only the integer value of moments is taken here because moments cannot be decimal
mazTRI(1.9,0.5)