dAddBin function

Additive Binomial Distribution

These functions provide the ability for generating probability function values and cumulative probability function values for the Additive Binomial Distribution.

dAddBin(x,n,p,alpha)

Arguments

• x: vector of binomial random variables.
• n: single value for no of binomial trials.
• p: single value for probability of success
• alpha: single value for alpha parameter.

Returns

The output of dAddBin gives a list format consisting

pdf probability function values in vector form.

mean mean of Additive Binomial Distribution.

var variance of Additive Binomial Distribution.

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.

$$P_{AddBin}(x)= {n \choose x} p^x (1-p)^{n-x}(\frac{alpha}{2}(\frac{x(x-1)}{p}+\frac{(n-x)(n-x-1)}{(1-p)}-\frac{alpha(n-1)n}{2})+1)$$

The alpha is in between

$$\frac{-2}{n(n-1)}min(\frac{p}{1-p},\frac{1-p}{p}) \le alpha \le (\frac{n+(2p-1)^2}{4p(1-p)})^{-1}$$ $$x = 0,1,2,3,...n$$ $$n = 1,2,3,...$$ $$0 < p < 1$$ $$-1 < alpha < 1$$

The mean and the variance are denoted as

$$E_{Addbin}[x]=np$$ $$Var_{Addbin}[x]=np(1-p)(1+(n-1)alpha)$$

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

Examples

#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="Additive 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,dAddBin(0:10,10,a[i],b[i])$pdf,col = col[i],lwd=2.85)
  points(0:10,dAddBin(0:10,10,a[i],b[i])$pdf,col = col[i],pch=16)
}

dAddBin(0:10,10,0.58,0.022)$pdf     #extracting the probability values
dAddBin(0:10,10,0.58,0.022)$mean    #extracting the mean
dAddBin(0:10,10,0.58,0.022)$var     #extracting the variance

#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="Additive 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,pAddBin(0:10,10,a[i],b[i]),col = col[i],lwd=2.85)
points(0:10,pAddBin(0:10,10,a[i],b[i]),col = col[i],pch=16)
}

pAddBin(0:10,10,0.58,0.022)       #acquiring the cumulative probability values

References

\insertRef johnson2005univariatefitODBOD

\insertRef kupper1978usefitODBOD

\insertRef paul1985threefitODBOD

\insertRef morel2012overdispersionfitODBOD