Distribution of $X_1+ X_2$ when $X_1, X_2, X_3$ follows a multinomial distribution

Question

The setting is that $X_1,X_2,X_3 \sim Multinomial(n;p_1,p_2,p_1+p_2)$. So the constraint is that $X_1+X_2+X_3=n$, and $p_1+p_2+(p_1+p_2) = 1$. I don't really understand how I should go about finding the distribution of $X_1+X_2$, could anyone help? Thanks.

Answer 1

From the constraint, we have that $2p_1 + 2p_2 = 1 \Rightarrow p_1 + p_2 = \frac{1}{2}$. Also the marginal distribution of each $X_i$ is marginally $Bin(n, p_i)$, as explained in the second answer here.

So, $$ P(X_1 + X_2 = t) = P(X_3 = n - t) = \frac{n!}{t!(n-t)!} (p_1 + p_2)^{n-t} (1-p_1 - p_2)^{t} = \frac{n!}{t!(n-t)!} \frac{1}{2^n{}} $$