# pTRI: Triangular Distribution bounded between [0,1] In Amalan-ConStat/R-fitODBOD: Modeling Over Dispersed Binomial Outcome Data Using BMD and ABD

## 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

 1 pTRI(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 ≤ p < c

g_{P}(p)= \frac{2(1-p)}{(1-c)}

; c ≤ p ≤ 1

G_{P}(p)= \frac{p^2}{c}

; 0 ≤ p < c

G_{P}(p)= 1-\frac{(1-p)^2}{(1-c)}

; c ≤ p ≤ 1

0 ≤ mode=c ≤ 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 pTRI gives the cumulative density values in vector form

## 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.

triangle

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Triangular

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 #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) 

Amalan-ConStat/R-fitODBOD documentation built on Oct. 1, 2018, 7:13 p.m.