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 \(Y\).

Answer 1

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.

Answer 2

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