These functions provide the ability for generating probability function values and cumulative probability function values for the Beta-Correlated Binomial Distribution.
Arguments
- x
vector of binomial random variables.
- n
single value for no of binomial trials.
- cov
single value for covariance.
- a
single value for alpha parameter.
- b
single value for beta parameter.
Value
The output of dBetaCorrBin gives a list format consisting
pdf probability function values in vector form.
mean mean of Beta-Correlated Binomial Distribution.
var variance of Beta-Correlated Binomial Distribution.
corr correlation of Beta-Correlated Binomial Distribution.
mincorr minimum correlation value possible.
maxcorr maximum correlation value possible.
Details
The probability function and cumulative function can be constructed and are denoted below
The cumulative probability function is the summation of probability function values.
$$x = 0,1,2,3,...n$$ $$n = 1,2,3,...$$ $$0 < a,b$$ $$-\infty < cov < +\infty $$ $$0 < p < 1$$
$$p=\frac{a}{a+b}$$ $$\Theta=\frac{1}{a+b}$$
The Correlation is in between $$\frac{-2}{n(n-1)} min(\frac{p}{1-p},\frac{1-p}{p}) \le correlation \le \frac{2p(1-p)}{(n-1)p(1-p)+0.25-fo} $$ where \(fo=min [(x-(n-1)p-0.5)^2] \)
The mean and the variance are denoted as $$E_{BetaCorrBin}[x]= np$$ $$Var_{BetaCorrBin}[x]= np(1-p)(n\Theta+1)(1+\Theta)^{-1}+n(n-1)cov$$ $$Corr_{BetaCorrBin}[x]=\frac{cov}{p(1-p)}$$
NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.
References
Paul SR (1985). “A three-parameter generalization of the binomial distribution.” History and Philosophy of Logic, 14(6), 1497--1506.
Examples
#plotting the random variables and probability values
col <- rainbow(5)
a <- c(9.0,10,11,12,13)
b <- c(8.0,8.1,8.2,8.3,8.4)
plot(0,0,main="Beta-Correlated 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,dBetaCorrBin(0:10,10,0.001,a[i],b[i])$pdf,col = col[i],lwd=2.85)
points(0:10,dBetaCorrBin(0:10,10,0.001,a[i],b[i])$pdf,col = col[i],pch=16)
}
dBetaCorrBin(0:10,10,0.001,10,13)$pdf #extracting the pdf values
#> [1] 0.010992133 0.047498928 0.108466364 0.170474118 0.202669472 0.189491674
#> [7] 0.140699312 0.081828892 0.035693121 0.010567136 0.001618852
dBetaCorrBin(0:10,10,0.001,10,13)$mean #extracting the mean
#> [1] 4.347826
dBetaCorrBin(0:10,10,0.001,10,13)$var #extracting the variance
#> [1] 3.469017
dBetaCorrBin(0:10,10,0.001,10,13)$corr #extracting the correlation
#> [1] 0.004069231
dBetaCorrBin(0:10,10,0.001,10,13)$mincorr #extracting the minimum correlation value
#> [1] -0.01709402
dBetaCorrBin(0:10,10,0.001,10,13)$maxcorr #extracting the maximum correlation value
#> [1] 0.2145215
#plotting the random variables and cumulative probability values
col <- rainbow(5)
a <- c(9.0,10,11,12,13)
b <- c(8.0,8.1,8.2,8.3,8.4)
plot(0,0,main="Beta-Correlated 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,pBetaCorrBin(0:10,10,0.001,a[i],b[i]),col = col[i],lwd=2.85)
points(0:10,pBetaCorrBin(0:10,10,0.001,a[i],b[i]),col = col[i],pch=16)
}
pBetaCorrBin(0:10,10,0.001,10,13) #acquiring the cumulative probability values
#> [1] 0.01099213 0.05849106 0.16695742 0.33743154 0.54010101 0.72959269
#> [7] 0.87029200 0.95212089 0.98781401 0.99838115 1.00000000