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.


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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.