Chapter 5.2, Problem 10E

When $\alpha$ and $\beta$ are integers and $0<p<1$, we have ${\displaystyle {\int }_0^p\frac{\Gamma \left(\alpha +\beta \right)}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{y}^{\alpha -1}{\left(1-y\right)}^{\beta -1}dy}={\displaystyle \underset{y=\alpha }{\overset{n}{\sum }}\left(\begin{array}{c}n\\ y\end{array}\right)p^y{\left(1-p\right)}^{n-y}}$

where $n=\alpha +\beta -1$.

Verify this formula when $\alpha =4$ and $\beta =3$.

Hint: Integrate the left member by parts several times