# dBETA: Beta Distribution 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 moment about zero values for the Beta Distribution bounded between [0,1]

## Usage

 1 dBETA(p,a,b) 

## Arguments

 p vector of probabilities a single value for shape parameter alpha representing as a b single value for shape parameter beta representing as b

## Details

The probability density function and cumulative density function of a unit bounded beta distribution with random variable P are given by

g_{P}(p)= \frac{p^{a-1}(1-p)^{b-1}}{B(a,b)}

; 0 ≤ p ≤ 1

G_{P}(p)= \frac{B_p(a,b)}{B(a,b)}

; 0 ≤ p ≤ 1

a,b > 0

The mean and the variance are denoted by

E[P]= \frac{a}{a+b}

var[P]= \frac{ab}{(a+b)^2(a+b+1)}

The moments about zero is denoted as

E[P^r]= ∏_{i=0}^{r-1} (\frac{a+i}{a+b+i})

r = 1,2,3,...

Defined as B_p(a,b)=\int^p_0 t^{a-1} (1-t)^{b-1}\,dt is incomplete beta integrals and B(a,b) is the beta function.

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further

## Value

The output of dBETA gives a list format consisting

pdf probability density values in vector form

mean mean of the beta distribution

var variance of the beta distribution

## References

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

Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.

Beta
  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) a<-c(1,2,5,10) plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values", xlim = c(0,1),ylim = c(0,4)) for (i in 1:4) { lines(seq(0,1,by=0.01),dBETA(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i]) } dBETA(seq(0,1,by=0.01),2,3)$pdf #extracting the pdf values dBETA(seq(0,1,by=0.01),2,3)$mean #extracting the mean dBETA(seq(0,1,by=0.01),2,3)$var #extracting the variance #plotting the random variables and cumulative probability values col<-rainbow(4) a<-c(1,2,5,10) 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),pBETA(seq(0,1,by=0.01),a[i],a[i]),col = col[i]) } pBETA(seq(0,1,by=0.01),2,3) #acquiring the cumulative probability values mazBETA(1.4,3,2) #acquiring the moment about zero values mazBETA(2,3,2)-mazBETA(1,3,2)^2 #acquiring the variance for a=3,b=2 #only the integer value of moments is taken here because moments cannot be decimal mazBETA(1.9,5.5,6)