Joint Distribution of $W=X^{2}$ and $X$. Where X is uniform distributed

Question

How can I obtain the Joint Distribution of $W=X^{2}$ and $X$. Where X is uniform distributed in $[0,1]$?

Any hint guys?

Many thanks!

What Ive been done so far:

$F_{X,W}(x,w) = P(X \leq x; W \leq w) = P(X \leq x; X^{2} \leq w) = P(X \leq x; \sqrt w \leq X \leq \sqrt w) = P(X \leq x; 0 \leq X \leq \sqrt w) = P(X \leq \sqrt w) $

Which is: $F_{X,W}(x,w) = \sqrt w$ , where $w \in (0,1]$

Is this right? Its weird because I found that $F_{W} = \sqrt w$

What Am I doing wrong? Did I make any mistake?

Answer 1

$P(X\leq x, X\leq \sqrt w)=\min \{x,\sqrt w\}$ provided $\min \{x,\sqrt w\} <1$. If $x<0$ or $w<0$ then $F_{X,W} (x,w)=0$. If $\min \{x,\sqrt w\} >1$ then $F_{X,W} (x,w)=1$.