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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?

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  • $\begingroup$ Your $f_{AC}$ is incorrect (assuming by "$AC$" you mean the product of the variables $A$ and $C.$) $\endgroup$
    – whuber
    Feb 8, 2021 at 14:19
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    $\begingroup$ @whuber Thank you... aah I think i did a fundamental mistake there $\endgroup$
    – MathMan
    Feb 8, 2021 at 14:22

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