dGAMMA: Gamma Distribution

In fitODBOD: Modeling Over Dispersed Binomial Outcome Data Using BMD and ABD

Gamma Distribution

Description

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

Usage

dGAMMA(p,c,l)

Arguments

p	vector of probabilities.
c	single value for shape parameter c.
l	single value for shape parameter l.

Details

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

g_{P}(p) = \frac{ c^l p^{c-1}}{γ(l)} [ln(1/p)]^{l-1}

; 0 ≤ p ≤ 1

G_{P}(p) = \frac{ Ig(l,cln(1/p))}{γ(l)}

; 0 ≤ p ≤ 1

l,c > 0

The mean the variance are denoted by

E[P] = (\frac{c}{c+1})^l

var[P] = (\frac{c}{c+2})^l - (\frac{c}{c+1})^{2l}

The moments about zero is denoted as

E[P^r]=(\frac{c}{c+r})^l

r = 1,2,3,...

Defined as γ(l) is the gamma function Defined as Ig(l,cln(1/p))= \int_0^{cln(1/p)} t^{l-1} e^{-t}dt is the Lower incomplete gamma function

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

Value

The output of dGAMMA gives a list format consisting

pdf probability density values in vector form.

mean mean of the Gamma distribution.

var variance of Gamma distribution.

References

Olshen, A. C. Transformations of the Pearson Type III Distribution. Ann. Math. Statist. 9 (1938), no. 3, 176–200.

See Also

GammaDist

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),dGAMMA(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
}

dGAMMA(seq(0,1,by=0.01),5,6)$pdf   #extracting the pdf values
dGAMMA(seq(0,1,by=0.01),5,6)$mean  #extracting the mean
dGAMMA(seq(0,1,by=0.01),5,6)$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),pGAMMA(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
}

pGAMMA(seq(0,1,by=0.01),5,6)   #acquiring the cumulative probability values
mazGAMMA(1.4,5,6)              #acquiring the moment about zero values
mazGAMMA(2,5,6)-mazGAMMA(1,5,6)^2 #acquiring the variance for a=5,b=6

#only the integer value of moments is taken here because moments cannot be decimal
mazGAMMA(1.9,5.5,6)

fitODBOD documentation built on Jan. 15, 2023, 5:11 p.m.