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.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/Ukgln.png)