Prove that $$ \sum_{k=0}^{x}{n\choose k} p^k (a-p)^{n-k} = (n-x){n\choose x}\int_{0}^{1-p}t^{n-x-1}(1-t)^{x}dt $$ (Hint : Integrate by parts or differentiate both sides with respect to $p$)
From the book "Statistical Inference"
Until now, I figured out it is enough to show $$ P[\mathrm{Bin} (n,p) \leq x] = P[ \mathrm{Beta} (n-x,x+1) \leq 1-p] $$
But it is hard to develop the above equation. Can anybody help?