dGAMMA: 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)

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.

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

$g_{P} (p) = \frac{c^{l} p^{c - 1}}{γ (l)} [l n (1 / p)]^{l - 1}$ ; $0 \leq p \leq 1$ $G_{P} (p) = \frac{I g (l, c l n (1 / p))}{γ (l)}$ ; $0 \leq p \leq 1$ $l, c > 0$

The mean the variance are denoted by $E [P] = (\frac{c}{c + 1})^{l}$ $v a r [P] = (\frac{c}{c + 2})^{l} - (\frac{c}{c + 1})^{2 l}$

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 $I g (l, c l n (1 / p)) = \int_{0}^{c l n (1 / p)} t^{l - 1} e^{- t} d t$ 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.

References

olshen1938transformationsfitODBOD

Examples

Run this code

#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)

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