Probability: joint finding the joint pdf

Let $X$~$UNIF(-1,1)$ and $Y$~$UNIF(0,\frac{1}{10})$ and $X,Y$ are independent random variables. Define $Z=X^2+Y$. Find the joint pdf of $X$ and $Y$.

Question

Probability: joint finding the joint pdf

Let $X$~$UNIF(-1,1)$ and $Y$~$UNIF(0,\frac{1}{10})$ and $X,Y$ are independent random variables. Define $Z=X^2+Y$. Find the joint pdf of $X$ and $Y$.

kirbykirby · Answer

I found 
$f_X(x)=\frac{1}{2},-1 \le x \le 1$

$f_Y(y)=10,0 \le y \le \frac{1}{10}$

Since they are independent, 
$$f(x,y)=f_X(x)*f_Y(y)$$ 
$$=\frac{1}{2}*10=5, -1 \le x \le 1, 0 \le y \le \frac{1}{10}$$

Is this all there is? It looks deceptively simple.

kirbykirby · Answer

(perhaps the Z is irrelevant for this questions... this is part of a multi-part question with a), b), c)...)