# Solved – Distribution of the sum of two independent Beta-Binomial variables

Consider two independent discrete random variables \$y_1\$ and \$y_2\$, both distributed with a Beta-Binomial distribution, with different number of successes \$n_1\$ and \$n_2\$ but the same parameters \$a\$ and \$b\$

\$ p(y_1|n_1,a,b) = {n_1 choose y_1} dfrac{B(y_1 + a,n_1 -y_1 +b)}{B(a,b)} \$

\$ p(y_2|n_2,a,b) = {n_2 choose y_2} dfrac{B(y_2 + a,n_2 -y_2 +b)}{B(a,b)} \$

Consider a discrete variable \$Z = y_1 + y_2\$. Is \$Z\$ distributed as well as a Beta-Binomial (with parameters \$n_1+n_2\$, \$a'\$ and \$b'\$?

I could not prove it in an analytic form so far, but I have been trying it out with some simulations, at least to check whether the assumption is wrong in some cases. Reparametrising \$a\$ and \$b\$ as \$mu = dfrac{a}{a+b}\$ and \$rho = dfrac{1}{a+b+1}\$, \$y_1\$ and \$y_2\$ have mean \$mu n_1\$ and \$mu n_2\$ respectively and variance \$mu(1-mu)n_1(1 + (n_1-1)rho)\$ and \$mu(1-mu)n_2(1 + (n_2-1)rho)\$ respectively.

Based on independence, the mean of \$Z\$ is \$mu(n_1+n_2)\$ and the variance of \$Z\$ is \$mu(1-mu)(n_1(1 + (n_1-1)rho) + n_2(1 + (n_2-1)rho))\$. If \$Z\$ was distributed according to a beta-binomial distribution, then it would have paramters \$mu'\$ and \$rho'\$, with \$mu'= mu\$ and

\$rho' = dfrac{dfrac{n_1(1 + (n_1-1)rho) + n_2(1 + (n_2-1)rho)}{n_1+n_2} – 1}{n_1+n_2-1} = rho dfrac{n_1(n_1-1)+n_2(n_2-1)}{(n_1+n_2)(n_1+n_2 -1)} \$

Here is some code to generate \$Z\$ as a sum of two independent Beta-Binomials (sorry about the code, R is not my main language)

``n1 = 20 n2 = 50 mu = .6 k  = 20  p1  = rbeta(1e6,mu*k,(1-mu)*k) y1  = rbinom(1e6,n1,p1)  p2  = rbeta(1e6,mu*k,(1-mu)*k) y2  = rbinom(1e6,n2,p2)  z   = y1+y2  rho  = 1/(k+1) rho1 = rho*(n1*(n1-1)+n2*(n2-1))/((n1+n2)*(n1+n2-1)) k1   = 1/ rho1 - 1 p3  = rbeta(1e6,mu*k1,(1-mu)*k1) z1  = rbinom(1e6,n1+n2,p3)  print(c(var(z),var(z1))) plot(density(z,width= 3)) lines(density(z1,width = 3)) ``

I have been trying this code for different values of \$n_1\$, \$n_2\$, \$mu\$ and \$k\$, but in all the cases the variances using the sum of two beta-binomials or an appropriately tuned beta-binomial are very similar (the densities look indistinguishable)

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