dTRI {fitODBOD} | R Documentation |
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
Horsnell G (1957). “Economical acceptance sampling schemes.” Journal of the Royal Statistical Society. Series A (General), 120(2), 148–201. Johnson NL, Kotz S, Balakrishnan N (1995). Continuous univariate distributions, volume 2, volume 289. John wiley and sons. Karlis D, Xekalaki E (2008). The polygonal distribution. Springer. Okagbue HI, Edeki SO, Opanuga AA, Oguntunde PE, Adeosun ME (2014). “Using the Average of the Extreme Values of a Triangular Distribution for a Transformation, and Its Approximant via the Continuous Uniform Distribution.” British Journal of Mathematics and Computer Science, 4(24), 3497.
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)