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Let X has a uniform distribution on the interval (0,1). Given that X=x, Y has a uniform distribution on the interval (0,x). Find the marginal p.d.f. of Y.
My thought: marginal pdf $f_Y(y)=\int_0^1f(x,y)dx=\int_0^1f_{Y|X=x}(y)f_X(x)dx=\int_0^1\frac{1}{x}dx$.But the last integral goes to negative infinity.What's wrong?Could anyone help?

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You forgot the fact that conditionally, $Y<x$ so $f_{Y\mid X}(y\mid x)$ only has support on $x>y,$ so you should have $$ \int_y^1 \frac{1}{x}dx.$$

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