# Expected value of X and Y, E[XY] for uniform random variables

1. Nov 10, 2014

### Dustinsfl

1. The problem statement, all variables and given/known data
If $X\sim\mathcal{U}(-1,1)$ and $Y = X^2$, is it possible to determine to $cov(X, Y)$?

2. Relevant equations
\begin{align}
f_x &=
\begin{cases}
1/2, & -1<x<1\\
0, & \text{otherwise}
\end{cases}\\
f_y &=
\begin{cases}
1/\sqrt{y}, & 0<x<1\\
0, & \text{otherwise}
\end{cases}
\end{align}
3. The attempt at a solution
$$cov(X,Y) = E[XY] - E[X]E[Y] = E[XY] - 0\cdot 1/2 = E[XY]$$
Now
$$E[XY] = \int_0^1\int_{-1}^1g(X, Y)f_{x,y}(x,y)dxdy$$
From the information that I have, can I determine $E[XY]$?

2. Nov 10, 2014

### Orodruin

Staff Emeritus
I suggest using the fact that $Y = X^2$...

3. Nov 10, 2014

### Dustinsfl

How?

4. Nov 10, 2014

### Orodruin

Staff Emeritus
What do you get if you replace $Y$ everywhere by $X^2$?

5. Nov 10, 2014

### Ray Vickson

For any observed value $X = x$, the observed (or, rather, computed) value of $Y$ is $y = x^2$. It not just that $Y$ and $X^2$ have the same distribution; much more than that is true: $Y$ IS $X^2$.