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Let X and Y be independent variables with densities f and g concentrated on $(0, \infty)$. If E(X) < $\infty$ , show that the ratio X/Y has a finite expectation iff

$$ \int_0^1 \frac{1}{y} g(y)dy < \infty $$

I know that I have show both sides. Can I just use the expectation formula for continuous variables

$$ \int x f(x) dx $$ for a density f(x) of the variable X?

Thanks!

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