I'm wondering about the following question:

Let the random variables $\displaystyle $N,X_1,X_2,...$ be independent, $\displaystyle $N \in Po(\lambda)$$ and $\displaystyle $X_k \in Be(1/2), k \ge 1$$. Set

$\displaystyle $Y_1= \sum_{k=1}^N X_k$$ and $\displaystyle $Y_2 = N - Y_1$$, $\displaystyle $Y_1=0$$ for $\displaystyle $N=0$$. Show that $\displaystyle $Y_1$$ and $\displaystyle $Y_2$$ are independent and determine their distributions.

My thought so far:

I see that $\displaystyle Y_1$ given $\displaystyle N=n$ is binomially distributed with parameters n and 1/2. What I want to show is that $\displaystyle $p_{Y_1,Y_2}(i,j)=p_{Y_1}(i) \cdot p_{Y_2}(j)$$ and one can expand

$\displaystyle $p_{Y_1,Y_2}(i,j)=Pr(Y_1=i,Y_2=j) = \sum_{n=0}^N Pr(Y_1=i,Y_2=j | N=n) \cdot Pr(N=n)$$

according to the law of total probability, but I'm kind of stuck after that.