beta distribution as ratio gamma distributions I need a proof of this statement please:
Let $Y_1$ and $Y_2$ be independent random variables, where $Y_1$ is gamma distributed with parameters $\alpha$ and 1 and $Y_2$ is gamma distributed with parameters $\beta$ and 1. Then the random variable $X$ presented by the following formula:
\begin{equation} \label{lemma5}
    \displaystyle X = \frac{Y_1}{Y_1+Y_2}
\end{equation}
is beta distributed with parameters $\alpha$ and $\beta$.

It is not difficult. I cannot show you the entire proof because you question is without your own work but this is a useful hint:
To simplify the notation, let me set $X,Y$ as the two independent Gamma rv and let's derive the law of
$$U=\frac{X}{X+Y}$$
The starting point is the following system
$$\begin{cases}
u=\frac{x}{x+y} \\
v=x
\end{cases}\rightarrow \begin{cases}
x=v \\
y=v\frac{1-u}{u}
\end{cases}$$
with Jacobian $|J|=\frac{v}{u^2}$
Substitute in $f_{XY}(x,y)$ and solve the integral in $dv$ finding your beta density. It is not difficult so show your works amending your question and, just in case, I will take you to the solution
