Lovinson Multiplicative Binomial Distribution

Source: R/LMultiBin.R

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

pLMBin(x,n,p,phi)

Arguments

x

vector of binomial random variables.
n

single value for no of binomial trials.
p

single value for probability of success.
phi

single value for phi.

Value

The output of pLMBin gives cumulative probability values in vector form.

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.

$$P_{LMBin}(x)= {n \choose x} p^x (1-p)^{n-x} \frac{(phi^{x(n-x)}}{f(p,phi,n)} $$

here \(f(p,phi,n)\) is $$f(p,phi,n)= \sum_{k=0}^{n} {n \choose k} p^k (1-p)^{n-k} (phi^{k(n-k)} )$$

$$x = 0,1,2,3,...n$$ $$n = 1,2,3,...$$ $$k = 0,1,2,...,n$$ $$0 < p < 1$$ $$0 < phi $$

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

References

Elamir, E.A., 2013. Multiplicative-Binomial Distribution: Some Results on Characterization, Inference and Random Data Generation. Journal of Statistical Theory and Applications, 12(1), pp.92-105.

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="Lovinson Multiplicative 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,dLMBin(0:10,10,a[i],1+b[i])$pdf,col = col[i],lwd=2.85)
points(0:10,dLMBin(0:10,10,a[i],1+b[i])$pdf,col = col[i],pch=16)
}

dLMBin(0:10,10,.58,10.022)$pdf   #extracting the pdf values
#>  [1] 6.364309e-29 8.964365e-19 5.657070e-11 2.106255e-05 5.123785e-02
#>  [6] 8.509524e-01 9.771209e-02 7.659947e-05 3.923411e-10 1.185630e-17
#> [11] 1.605234e-27
dLMBin(0:10,10,.58,10.022)$mean   #extracting the mean
#> [1] 5.046585
dLMBin(0:10,10,.58,10.022)$var   #extracting the variance
#> [1] 0.1471704

#plotting 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="Lovinson Multiplicative 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,pLMBin(0:10,10,a[i],1+b[i]),col = col[i],lwd=2.85)
points(0:10,pLMBin(0:10,10,a[i],1+b[i]),col = col[i],pch=16)
}

pLMBin(0:10,10,.58,10.022)     #acquiring the cumulative probability values
#>  [1] 6.364309e-29 8.964365e-19 5.657070e-11 2.106261e-05 5.125891e-02
#>  [6] 9.022113e-01 9.999234e-01 1.000000e+00 1.000000e+00 1.000000e+00
#> [11] 1.000000e+00