fitODBOD (version 1.5.3)

dBETA: Beta Distribution

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

dBETA(p,a,b)

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.

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

$$g_{P}(p)= \frac{p^{a-1}(1-p)^{b-1}}{B(a,b)} $$ ; \(0 \le p \le 1\) $$G_{P}(p)= \frac{B_p(a,b)}{B(a,b)} $$ ; \(0 \le p \le 1\) $$a,b > 0$$

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 $$E[P^r]= \prod_{i=0}^{r-1} (\frac{a+i}{a+b+i}) $$ \(r = 1,2,3,...\)

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.

References

johnson1995continuousfitODBOD trenkler1996continuousfitODBOD

See Also

Beta

or

https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html

Examples

#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)