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.

Value

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.

References

Horsnell, G. (1957). Economic acceptance sampling schemes. Journal of the Royal Statistical Society, Series A, 120:148-191.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, Vol. 2, Wiley Series in Probability and Mathematical Statistics, Wiley.

Karlis, D. & Xekalaki, E., 2008. The Polygonal Distribution. In Advances in Mathematical and Statistical Modeling. Boston: Birkhuser Boston, pp. 21-33.

Available at: http://dx.doi.org/10.1007/978-0-8176-4626-4_2 .

Okagbue, H. et al., 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 & Computer Science, 4(24), pp.3497-3507.

Available at: http://www.sciencedomain.org/abstract.php?iid=699&id=6&aid=6427 .

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)$pdf     #extracting the pdf values
#>  [1] 0.0000000 0.3333333 0.6666667 1.0000000 1.3333333 1.6666667 2.0000000
#>  [8] 1.8571429 1.7142857 1.5714286 1.4285714 1.2857143 1.1428571 1.0000000
#> [15] 0.8571429 0.7142857 0.5714286 0.4285714 0.2857143 0.1428571 0.0000000
dTRI(seq(0,1,by=0.01),0.3)$mean    #extracting the mean
#> [1] 0.4333333
dTRI(seq(0,1,by=0.01),0.3)$var     #extracting the variance
#> [1] 0.04388889

#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
#>  [1] 0.000000000 0.008333333 0.033333333 0.075000000 0.133333333 0.208333333
#>  [7] 0.300000000 0.396428571 0.485714286 0.567857143 0.642857143 0.710714286
#> [13] 0.771428571 0.825000000 0.871428571 0.910714286 0.942857143 0.967857143
#> [19] 0.985714286 0.996428571 1.000000000
mazTRI(1.4,.3)                  #acquiring the moment about zero values
#> [1] 0.4333333
mazTRI(2,.3)-mazTRI(1,.3)^2     #variance for when is mode 0.3
#> [1] 0.04388889

#only the integer value of moments is taken here because moments cannot be decimal
mazTRI(1.9,0.5)
#> [1] 0.5