How can we find the distribution function of an Uniform Random variable with Random variable bounds?

Question

X is a uniform random variable in (0,1) and Y is a uniform random variable in (X,1). How can I find the probability density function of Y? I thought and searched a lot and I found nothing. please help me.

Answer 1

Hint:

$P(Y \le y \mid X=x)= \dfrac{y-x}{1-x}$ provided that $x \le y$

So $\displaystyle P(Y \le y) = \int_{x=0}^{x=y} \frac{y-x}{1-x} \,dx$

Then differentiate with respect to $y$ to get the density

Answer 2

$$f_Y(y)=\int f_{(X,Y)}(x,y)dx=\int f_{Y|X=x}(y)f_X(x)dx$$

You know $f_X$ and you know $f_{Y|X=x}$.