Suppose that the random variable $X$ is uniformly distributed symmetrically around zero, but in such a way that the parameter is uniform on $(0,1)$; that is, suppose that $$X\mid A=a \in U (-a,a),\qquad A \in U (0,1)$$ Find the distribution of $X$.
This is my approach:
$$F_{X}(x)=P(X\leq x)=\int_{\Re}P (X\leq x \mid A=a)\cdot f_{A}(a)\cdot da=\int_{0}^{1}P (X\leq x \mid A=a)\cdot da$$
Next, I determine $P (X\leq x \mid A=a)$ and got the following:
$$P (X\leq x \mid A=a)= \left\{ \begin{array}{ll} 0 & x\leq -a \\ \frac{x+a}{2a} & -a\leq x\leq a \\ 1 & a\leq x \\ \end{array} \right. $$
It is from here on that I have problems. I don't know how to properly think about the support of $a$ when evaluating $$\int_{0}^{1}P (X\leq x \mid A=a)\cdot da.$$