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].
dKUM(p,a,b)
p | vector of probabilities. |
---|---|
a | single value for shape parameter alpha representing as a. |
b | single value for shape parameter beta representing as b. |
The output of dKUM
gives a list format consisting
pdf
probability density values in vector form.
mean
mean of the Kumaraswamy distribution.
var
variance of the Kumaraswamy distribution.
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.
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.
#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))#> [1] 0.000000000 0.059988001 0.119904019 0.179676146 0.239232614 0.298501875 #> [7] 0.357412666 0.415894084 0.473875661 0.531287429 0.588060000 0.644124631 #> [13] 0.699413299 0.753858776 0.807394694 0.859955625 0.911477146 0.961895914 #> [19] 1.011149741 1.059177659 1.105920000 1.151318461 1.195316179 1.237857806 #> [25] 1.278889574 1.318359375 1.356216826 1.392413344 1.426902221 1.459638689 #> [31] 1.490580000 1.519685491 1.546916659 1.572237236 1.595613254 1.617013125 #> [37] 1.636407706 1.653770374 1.669077101 1.682306519 1.693440000 1.702461721 #> [43] 1.709358739 1.714121066 1.716741734 1.717216875 1.715545786 1.711731004 #> [49] 1.705778381 1.697697149 1.687500000 1.675203151 1.660826419 1.644393296 #> [55] 1.625931014 1.605470625 1.583047066 1.558699234 1.532470061 1.504406579 #> [61] 1.474560000 1.442985781 1.409743699 1.374897926 1.338517094 1.300674375 #> [67] 1.261447546 1.220919064 1.179176141 1.136310809 1.092420000 1.047605611 #> [73] 1.001974579 0.955638956 0.908715974 0.861328125 0.813603226 0.765674494 #> [79] 0.717680621 0.669765839 0.622080000 0.574778641 0.528023059 0.481980386 #> [85] 0.436823654 0.392731875 0.349890106 0.308489524 0.268727501 0.230807669 #> [91] 0.194940000 0.161340871 0.130233139 0.101846216 0.076416134 0.054185625 #> [97] 0.035404186 0.020328154 0.009220781 0.002352299 0.000000000#> [1] 0.4571429#> [1] 0.04102041#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))#> [1] 0.000000000 0.000299970 0.001199520 0.002697571 0.004792324 0.007481266 #> [7] 0.010761167 0.014628088 0.019077382 0.024103701 0.029701000 0.035862542 #> [13] 0.042580906 0.049847997 0.057655050 0.065992641 0.074850697 0.084218508 #> [19] 0.094084732 0.104437416 0.115264000 0.126551336 0.138285700 0.150452806 #> [25] 0.163037823 0.176025391 0.189399636 0.203144190 0.217242210 0.231676393 #> [31] 0.246429000 0.261481874 0.276816462 0.292413838 0.308254724 0.324319516 #> [37] 0.340588302 0.357040896 0.373656856 0.390415514 0.407296000 0.424277274 #> [43] 0.441338152 0.458457333 0.475613434 0.492785016 0.509950617 0.527088785 #> [49] 0.544178110 0.561197257 0.578125000 0.594940258 0.611622130 0.628149931 #> [55] 0.644503231 0.660661891 0.676606099 0.692316417 0.707773813 0.722959704 #> [61] 0.737856000 0.752445144 0.766710156 0.780634672 0.794202997 0.807400141 #> [67] 0.820211870 0.832624752 0.844626203 0.856204533 0.867349000 0.878049854 #> [73] 0.888298390 0.898086996 0.907409210 0.916259766 0.924634649 0.932531150 #> [79] 0.939947921 0.946885026 0.953344000 0.959327906 0.964841391 0.969890743 #> [85] 0.974483952 0.978630766 0.982342755 0.985633371 0.988518007 0.991014061 #> [91] 0.993141000 0.994920422 0.996376121 0.997534153 0.998422901 0.999073141 #> [97] 0.999518110 0.999793575 0.999937901 0.999992119 1.000000000#> [1] 0.6428571#> [1] 0.04102041#only the integer value of moments is taken here because moments cannot be decimal mazKUM(1.9,5.5,6)#> [1] 0.6551253