dTRI: Triangular Distribution Bounded Between [0,1]

In fitODBOD: Modeling Over Dispersed Binomial Outcome Data Using BMD and ABD

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)

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.

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

References

\insertRef

horsnell1957economicalfitODBOD \insertRefjohnson1995continuousfitODBOD \insertRefkarlis2008polygonalfitODBOD \insertRefokagbue2014usingfitODBOD

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)

fitODBOD documentation built on Oct. 10, 2024, 5:07 p.m.