These functions provide the ability for generating probability function values and cumulative probability function values for the COM Poisson Binomial Distribution.
dCOMPBin(x,n,p,v)
x | vector of binomial random variables. |
---|---|
n | single value for no of binomial trials. |
p | single value for probability of success. |
v | single value for v. |
The output of dCOMPBin
gives a list format consisting
pdf
probability function values in vector form.
mean
mean of COM Poisson Binomial Distribution.
var
variance of COM Poisson Binomial Distribution.
The probability function and cumulative function can be constructed and are denoted below
The cumulative probability function is the summation of probability function values.
$$P_{COMPBin}(x) = \frac{{n \choose x}^v p^x (1-p)^{n-x}}{\sum_{j=0}^{n} {n \choose j}^v p^j (1-p)^{(n-j)}}$$ $$x = 0,1,2,3,...n$$ $$n = 1,2,3,...$$ $$0 < p < 1$$ $$-\infty < v < +\infty $$
NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.
Extracted from
Borges, P., Rodrigues, J., Balakrishnan, N. and Bazan, J., 2014. A COM-Poisson type generalization of the binomial distribution and its properties and applications. Statistics & Probability Letters, 87, pp.158-166.
Available at: http://conteudo.icmc.usp.br/CMS/Arquivos/arquivos_enviados/BIBLIOTECA_113_NSE_90.pdf
#plotting the random variables and probability values col <- rainbow(5) a <- c(0.58,0.59,0.6,0.61,0.62) b <- c(0.022,0.023,0.024,0.025,0.026) plot(0,0,main="COM Poisson Binomial probability function graph",xlab="Binomial random variable", ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))for (i in 1:5) { lines(0:10,dCOMPBin(0:10,10,a[i],b[i])$pdf,col = col[i],lwd=2.85) points(0:10,dCOMPBin(0:10,10,a[i],b[i])$pdf,col = col[i],pch=16) }dCOMPBin(0:10,10,0.58,0.022)$pdf #extracting the pdf values#> [1] 0.01058483 0.01537669 0.02194887 0.03097149 0.04329998 0.06003554 #> [7] 0.08257434 0.11263594 0.15222443 0.20337258 0.26697531dCOMPBin(0:10,10,0.58,0.022)$mean #extracting the mean#> [1] 7.627366dCOMPBin(0:10,10,0.58,0.022)$var #extracting the variance#> [1] 5.803804#plotting the random variables and cumulative probability values col <- rainbow(5) a <- c(0.58,0.59,0.6,0.61,0.62) b <- c(0.022,0.023,0.024,0.025,0.026) plot(0,0,main="COM Poisson Binomial probability function graph",xlab="Binomial random variable", ylab="Probability function values",xlim = c(0,10),ylim = c(0,1))for (i in 1:5) { lines(0:10,pCOMPBin(0:10,10,a[i],b[i]),col = col[i],lwd=2.85) points(0:10,pCOMPBin(0:10,10,a[i],b[i]),col = col[i],pch=16) }#> [1] 0.01058483 0.02596152 0.04791039 0.07888188 0.12218187 0.18221740 #> [7] 0.26479174 0.37742768 0.52965211 0.73302469 1.00000000