pTRI function

Triangular Distribution Bounded Between [0,1]

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].

pTRI(p,mode)

Arguments

• p: vector of probabilities.
• mode: single value for mode.

Returns

The output of pTRI gives the cumulative density values in vector form.

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.

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)

References

\insertRef horsnell1957economicalfitODBOD

\insertRef johnson1995continuousfitODBOD

\insertRef karlis2008polygonalfitODBOD

\insertRef okagbue2014usingfitODBOD