fitODBOD (version 1.5.3)

dBetaBin: Beta-Binomial Distribution

Description

These functions provide the ability for generating probability function values and cumulative probability function values for the Beta-Binomial Distribution.

Usage

dBetaBin(x,n,a,b)

Value

The output of dBetaBin gives a list format consisting

pdf probability function values in vector form.

mean mean of the Beta-Binomial Distribution.

var variance of the Beta-Binomial Distribution.

over.dis.para over dispersion value of the Beta-Binomial Distribution.

Arguments

x

vector of binomial random variables.

n

single value for no of binomial trials.

a

single value for shape parameter alpha representing as a.

b

single value for shape parameter beta representing as b.

Details

Mixing Beta distribution with Binomial distribution will create the Beta-Binomial distribution. The probability function and cumulative probability function can be constructed and are denoted below.

The cumulative probability function is the summation of probability function values.

$$P_{BetaBin}(x)= {n \choose x} \frac{B(a+x,n+b-x)}{B(a,b)} $$ $$a,b > 0$$ $$x = 0,1,2,3,...n$$ $$n = 1,2,3,...$$

The mean, variance and over dispersion are denoted as $$E_{BetaBin}[x]= \frac{na}{a+b} $$ $$Var_{BetaBin}[x]= \frac{(nab)}{(a+b)^2} \frac{(a+b+n)}{(a+b+1)} $$ $$over dispersion= \frac{1}{a+b+1} $$

Defined as B(a,b) is the beta function.

References

young2008poolingfitODBOD trenkler1996continuousfitODBOD hughes1993usingfitODBOD

Examples

#plotting the random variables and probability values
col <- rainbow(5)
a <- c(1,2,5,10,0.2)
plot(0,0,main="Beta-binomial probability function graph",xlab="Binomial random variable",
ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))
for (i in 1:5)
{
lines(0:10,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],lwd=2.85)
points(0:10,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],pch=16)
}

dBetaBin(0:10,10,4,.2)$pdf    #extracting the pdf values
dBetaBin(0:10,10,4,.2)$mean   #extracting the mean
dBetaBin(0:10,10,4,.2)$var    #extracting the variance
dBetaBin(0:10,10,4,.2)$over.dis.para  #extracting the over dispersion value

#plotting the random variables and cumulative probability values
col <- rainbow(4)
a <- c(1,2,5,10)
plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable",
ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1))
for (i in 1:4)
{
lines(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
points(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
}

pBetaBin(0:10,10,4,.2)   #acquiring the cumulative probability values