Show that the random variables $Y_1$ and $Y_2$ are independent 
Let $X_1,X_2$ be i.i.d with pdf
  $$f_X(x)=\begin{cases}
e^{-x} & \text{for } 0< x<\infty{}
\\0 & \text{elsewhere }
\end{cases}$$
  Show that the random variables $Y_1$ and $Y_2$ with $Y_1=X_1+X_2$ and $Y_2=\frac{X_1}{X_1+X_2}$ are independent.

I know that for $Y_1$ and $Y_2$ to be independent. $P(Y_1\cap Y_2)=P(Y_1)P(Y_2)$.

Here is a simulation of 100,000 $(Y_1, Y_2)$-pairs from R statistical software.
The $X_i$ are iid $Exp(rate=1),$ $Y_1 \sim Gamma(shape=2, rate=1)$
and $Y_2 \sim Unif(0, 1).$ Also, $Y_1$ and $Y_2$ are uncorrelated. (If these
distributions are not covered in your text, you can see Wikipedia articles
on 'exponential distribution' and 'gamma distribution'.)
x1 = rexp(10^5);  x2 = rexp(10^5)
y1 = x1 + x2;  y2 = x1/y1
cor(y1,y2)
## 0.002440974 # consistent with 0 population correlation

In the figure below, the first panel shows no pattern of association
between $Y_1$ and $Y_2$. Of course, this is no formal proof of independence, but if you
do the bivariate transformation to get the joint density function of
$Y_1$ and $Y_2,$ you should be able to see that it factors into the
PDFs of $Y_1$ and $Y_2$. These PDFs are plotted along with the histograms
of the simulated distributions of $Y_1$ and $Y_2$ in the second and third
panels, respectively.

