fitODBOD (version 1.4.1-1)

mazKUM: Kumaraswamy Distribution

Description

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

Usage

mazKUM(r,a,b)

Value

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

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.

Details

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

$$g_{P}(p)= abp^{a-1}(1-p^a)^{b-1} $$ ; \(0 \le p \le 1\) $$G_{P}(p)= 1-(1-p^a)^b$$ ; \(0 \le p \le 1\) $$a,b > 0$$

The mean and the variance are denoted by $$E[P]= bB(1+\frac{1}{a},b)$$ $$var[P]= bB(1+\frac{2}{a},b)-(bB(1+\frac{1}{a},b))^2$$

The moments about zero is denoted as $$E[P^r]= bB(1+\frac{r}{a},b)$$ \(r = 1,2,3,...\)

Defined as \(B(a,b)\) is the beta function.

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

References

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1), 79-88.

Available at : http://dx.doi.org/10.1016/0022-1694(80)90036-0.

Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(1), 70-81.

Available at : http://dx.doi.org/10.1016/j.stamet.2008.04.001.

See Also

Kumaraswamy

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(0,6))
for (i in 1:4)
{
lines(seq(0,1,by=0.01),dKUM(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
}

dKUM(seq(0,1,by=0.01),2,3)$pdf   #extracting the probability values
dKUM(seq(0,1,by=0.01),2,3)$mean  #extracting the mean
dKUM(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),pKUM(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
}

pKUM(seq(0,1,by=0.01),2,3)    #acquiring the cumulative probability values
mazKUM(1.4,3,2)               #acquiring the moment about zero values
mazKUM(2,2,3)-mazKUM(1,2,3)^2  #acquiring the variance for a=2,b=3

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