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

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
