mazBETA() R function from [fitODBOD]

Beta Distribution

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].


mazBETA(r,a,b)

Arguments

r: vector of moments.
a: single value for shape parameter alpha representing as a.
b: single value for shape parameter beta representing as b.

Returns

The output of mazBETA gives the moments about zero in vector form.

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.

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)

References

\insertRef johnson1995continuousfitODBOD

\insertRef trenkler1996continuousfitODBOD

mazBETA function