dGBeta1 function

Generalized Beta Type-1 Distribution

These functions provide the ability for generating probability density values, cumulative probability density values and moment about zero values for the Generalized Beta Type-1 Distribution bounded between [0,1].

dGBeta1(p,a,b,c)

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.
• c: single value for shape parameter gamma representing as c.

Returns

The output of dGBeta1 gives a list format consisting

pdf probability density values in vector form.

mean mean of the Generalized Beta Type-1 Distribution.

var variance of the Generalized Beta Type-1 Distribution.

Details

The probability density function and cumulative density function of a unit bounded Generalized Beta Type-1 Distribution with random variable P are given by

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

; $0 \le p \le 1$

$$G_{P}(p)= \frac{p^{ac}}{aB(a,b)} 2F1(a,1-b;p^c;a+1)$$

$0 \le p \le 1$

$$a,b,c > 0$$

The mean and the variance are denoted by

$$E[P]= \frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})}$$ $$var[P]= \frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2$$

The moments about zero is denoted as

$$E[P^r]= \frac{B(a+b,\frac{r}{c})}{B(a,\frac{r}{c})}$$

$r = 1,2,3,....$

Defined as $B(a,b)$ is Beta function. Defined as $2F1(a,b;c;d)$ is Gaussian Hypergeometric 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(5)
a <- c(.1,.2,.3,1.5,2.15)
plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
xlim = c(0,1),ylim = c(0,10))
for (i in 1:5)
{
lines(seq(0,1,by=0.001),dGBeta1(seq(0,1,by=0.001),a[i],1,2*a[i])$pdf,col = col[i])
}

dGBeta1(seq(0,1,by=0.01),2,3,1)$pdf    #extracting the pdf values
dGBeta1(seq(0,1,by=0.01),2,3,1)$mean   #extracting the mean
dGBeta1(seq(0,1,by=0.01),2,3,1)$var    #extracting the variance

pGBeta1(0.04,2,3,4)        #acquiring the cdf values for a=2,b=3,c=4
mazGBeta1(1.4,3,2,2)              #acquiring the moment about zero values
mazGBeta1(2,3,2,2)-mazGBeta1(1,3,2,2)^2        #acquiring the variance for a=3,b=2,c=2

#only the integer value of moments is taken here because moments cannot be decimal
mazGBeta1(3.2,3,2,2)

References

\insertRef manoj2013mcdonaldfitODBOD

\insertRef janiffer2014estimatingfitODBOD

\insertRef roozegar2017mcdonaldfitODBOD