I want to evaluate the following summation:
$$\sum^{n-1}_{k=0}{n\choose k}x^k(1-x)^{n-k-1}\sum^{n-1}_{j=k}{n-1\choose j}y^j(1-y)^{n-j-1}.$$ Both $x$ and $y$ are some numbers in $[0,1]$. Each one is just sum of binomial probabilities with success probability $x$ and $y$, but I can't figure out how to solve out this summation when there are two together..
I tired to use things similar to Vandermonde's identity, but I haven't been successful yet.. Can anyone provide any suggestion?
One thing I could do is representing the terms related to $y$ into an incomplete beta function. Since $$\sum^{n-1}_{j=k}{n-1\choose j}y^j(1-y)^{n-1-j}=1-I_{1-y}(n-k,k)=I_{y}(k,n-k),$$ the summation further simplified as $$\sum^{n-1}_{k=0}{n\choose k}x^k(1-x)^{n-k-1}I_y(k,n-k).$$ Would any further simplification be possible from this point?