mazTRI {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
mazTRI(r,mode)
Arguments
r |
vector of moments. |
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 mazTRI
give the moments about zero in vector form.
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)