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This is a question from exam review sheet. Please give me some guidance here. I do not know how can I find fY(y) without having information on f(x,y) or at least fX(x)?

Consider a random variable Y generated as follows. First select a value of X = x at random (uniform) over the interval (0,1). Then select a value of Y = y at random (uniform) over the interval (0,x). Find the probability density function fY(y).

Thank you.

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You know $X\sim U(0,1)$. Further, you know conditional on $X=x$, that $Y\sim U(0,x)$. That is the conditional density of $Y$ given $X=x$ is $f(y|x)=\frac{1}{x} \mathbb{1}_{0<y<x}$.

The unconditional density can then by found by the law of total probability, i.e. $$f_Y(y)=\int_0^1 f(y| x)f_X(x)dx,$$

Can you finish up from this?

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