pBetaBin | R Documentation |
These functions provide the ability for generating probability function values and cumulative probability function values for the Beta-Binomial Distribution.
pBetaBin(x,n,a,b)
x |
vector of binomial random variables. |
n |
single value for no of binomial trials. |
a |
single value for shape parameter alpha representing as a. |
b |
single value for shape parameter beta representing as b. |
Mixing Beta distribution with Binomial distribution will create the Beta-Binomial distribution. The probability function and cumulative probability function can be constructed and are denoted below.
The cumulative probability function is the summation of probability function values.
P_{BetaBin}(x)= {n \choose x} \frac{B(a+x,n+b-x)}{B(a,b)}
a,b > 0
x = 0,1,2,3,...n
n = 1,2,3,...
The mean, variance and over dispersion are denoted as
E_{BetaBin}[x]= \frac{na}{a+b}
Var_{BetaBin}[x]= \frac{(nab)}{(a+b)^2} \frac{(a+b+n)}{(a+b+1)}
over dispersion= \frac{1}{a+b+1}
Defined as B(a,b)
is the beta function.
The output of pBetaBin
gives cumulative probability values in vector form.
Young-Xu, Y. & Chan, K.A., 2008. Pooling overdispersed binomial data to estimate event rate. BMC medical research methodology, 8(1), p.58.
Available at: doi: 10.1186/1471-2288-8-58.
Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
Available at: doi: 10.1016/0167-9473(96)90015-8.
Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease Incidence. Phytopathology, 83(9), p.759.
Available at: doi: 10.1094/PHYTO-83-759
#plotting the random variables and probability values col <- rainbow(5) a <- c(1,2,5,10,0.2) plot(0,0,main="Beta-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,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],lwd=2.85) points(0:10,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],pch=16) } dBetaBin(0:10,10,4,.2)$pdf #extracting the pdf values dBetaBin(0:10,10,4,.2)$mean #extracting the mean dBetaBin(0:10,10,4,.2)$var #extracting the variance dBetaBin(0:10,10,4,.2)$over.dis.para #extracting the over dispersion value #plotting the random variables and cumulative probability values col <- rainbow(4) a <- c(1,2,5,10) plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable", ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1)) for (i in 1:4) { lines(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i]) points(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i]) } pBetaBin(0:10,10,4,.2) #acquiring the cumulative probability values
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