Uniform Random variables If $X$ and $Y $are both uniform random variables on $[\ 0,1]$
How do I go about finding the $P(X>Y>c)$ for some c in $[\ 0,1]$?
hints appreciated
My idea was to find the probability of intersection of the two, but the first probability i had no solid ideas how to calculate
thanks
Edit: they are independent
 A: I assume $X,Y$ are independent.
$P(X^2 > Y > c) = \int_c^1 P(X^2> x) dx = \int_c^1 P(X> \sqrt{x}) dx = \int_c^1 1- \sqrt{x}dx$
$= 1-c - \int_c^1 \sqrt{x}dx = 1-c - \dfrac{2x^{3/2}}{3}|_{x=c}^{x=1} = 1 - c - (\dfrac{2}{3} - \dfrac{2c^{3/2}}{3}) = \dfrac{1}{3} - c +  \dfrac{2c^{3/2}}{3}.$
In the first inequality I condition on the value of $Y \in (c,1)$.
A: If $X$ and $Y$ are independent then $P\left[X^{2}>Y>c\right]=\int_{0}^{1}\int_{0}^{1}u\left(x,y\right)dxdy$
where $u\left(x,y\right)$ is defined by $\left(x,y\right)\mapsto1$
if $x^{2}>y>c$ and $\left(x,y\right)\mapsto0$ otherwise. This leads
to $$P\left[X^{2}>Y>c\right]=\int_{c}^{1}\int_{y^{\frac{1}{2}}}^{1}dxdy=\int_{c}^{1}1-y^{\frac{1}{2}}dy=y-\frac{2}{3}y^{\frac{3}{2}}|_{c}^{1}=\frac{1}{3}-c+\frac{2}{3}c^{\frac{3}{2}}$$
A: In the $x$-$y$ plane, draw a picture of the square with corners $(0,0)$, $(1,0)$, $(1,1)$, and $(0,1)$. That is where the joint density function "lives."
Draw the line $y=c$. 
Draw the parabola $y=x^2$. 
Let $A$ be the part of the square above the line $y=c$, and below the parabola $y=x^2$. 
We want the integral over $A$ of the joint density function, or more simply the area of $A$. 
The region $A$ is a little sort of triangular-looking region in the upper right corner of the square. 
Find its area by integration. Integrate first with respect to $y$, then with respect to $x$. (It doesn't really matter.) The picture will give you the limits of integration.  
Remark: We assumed the joint density function is $1$ on the square, or equivalently that $X$ and $Y$ are independent. But this should be specified in the statement of the problem. If $X$ and $Y$ are not independent, then we cannot compute the probability unless we are given additional information. 
