mazGAMMA() R function from [fitODBOD]

Gamma Distribution

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].


mazGAMMA(r,c,l)

Arguments

r: vector of moments.
c: single value for shape parameter c.
l: single value for shape parameter l.

Returns

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

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}}{\gamma(l)} [ln(1/p)]^{l-1}

; $0 \le p \le 1$

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

; $0 \le p \le 1$

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 $\gamma(l)$ 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.

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)

References

\insertRef olshen1938transformationsfitODBOD

mazGAMMA function