Suppose $A$ and $C$ are uniformly distributed over $(0,1)$ and are independent random variables. Then, I found the pdf of $AC$ using this method:
$$f_{AC} (a,c) = f_A (a) f_C(c) = \begin{cases} 1 & 0 \le a,c \le 1 \\ 0 & {\text{otherwise}} \end{cases} $$
However, if I try to find it in the following manner:
Suppose $K = AC \implies F_K(k) = \int~ _{K \le d} \int f_K(k)~dK = \int~_{AC \le k} \int 1 \cdot dA ~dC$
$$=\int_0^k \int_0^1 dC ~dA ~+~\int_k^1 \int_0^{\frac{k}{a}} dC~dA = k - \ln k$$
Differentiating , we get $f_K(k) = - \ln k$
Are both these answers right?
