Question

if X~B(n₁ = 5, p = 1/4) and Y~B(n = 8, p = 1/4).Then what is the distribution of X+Y?

Answer 1

If $X \sim B(n, p)$ and $Y \sim B(m, p)$ are independent binomial variables with the same probability $p,$ then $X + Y$  is again a binomial variable; its distribution is $Z=X+Y \sim B(n+m, p).$

Given $n_1=5, n=8, p=1/4.$

Then $Z=X+Y \sim B(n_1+n=13, p=1/4).$