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Let $A$ and $B$ be identical, independent random variables with probability density functions (PDF) $f_A(a)$ and $f_B(b)$. Let $C = AB$. Why is the PDF of $C$ not $f_A(a)f_B(b)$?

Aren't $A$ and $B$ independent? Apparently, I'm supposed to not have the variables $a$ and $b$ in the PDF of $C$. Why is that? What is the right way to find the PDF of $C$?

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    Are $f_A$ and $f_B$ explicitly given? If so, one way is to compute the cumulative distribution function of $C$ and then differentiate. For the cdf, we need $\Pr(C\le w)$. To calculate, we want to integrate the joint. density (product) over the region where $ab\le w$. Evaluating the double integral is not strictly necessary, since you can differentiate under the integral sign.2012-12-09
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    The _argument_ of $f_C$ should be a _single_ variable, say, $c$, no? Then, in your hypothesized identity $f_C(c) = f_A(a)f_B(b)$, what is the relationship between $a$, $b$, and $c$? Well, you might say, it really is $f_C(c) = f_A(c)f_B(c)$. Is that what you meant to ask?2012-12-09

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You could get $C=12$ from $A=3$ and $B=4$ or from $A=2$ and $B=6$ or a multitude of other possibilities.

So (assuming $A$ and $B$ are independent) $$f_C(c)=\int f_A(a) f_B(c/a) \, da.$$