Suppose that $R^2 = X^2 + Y^2$. Find $f_{R^2}$ and $f_R$ if
$f_X(x) = f_Y(y) =
\begin{cases}
1/2 &\text{if}\, -1 \leq x \leq 1\\
0 &\text{otherwise}
\end{cases}$
Since we are not given the distribution I know we have to use convolutions. Namely,
$$f_{R^2}(x) = \int_{-\infty}^{+\infty} f_{X^2}(t)f_{Y^2}(x-t)dt$$
My problem is that I was given $f_X(x)$ and $f_Y(y)$ and not $f_{X^2}(x)$ and $f_{Y^2}(y)$.
I'd appreciate any help. Thanks.
You can get the distribution of $X^2$ from the distribution for $X.$ For $l\in[0,1],$ the CDF of $X^2$ is
$$F_{X^2}(l) =P(X^2<l) = P(-\sqrt{l}<X<\sqrt{l}) =\sqrt l.$$ The PDF is the derivative of the CDF, so $$ f_{X^2}(l) = \frac{1}{2\sqrt{l}}$$ for $l\in [0,1]$ (and zero outside that support).