Originally Posted by

**notgoodatmath** Hi,

I have two random variables, $\displaystyle X_1$ and $\displaystyle X_2$ that have joint pdf:

$\displaystyle f_{(x_1,x_2)}(x_1,x_2) = \left\{

\begin{array}{c l}

2 & 0 \le x_1 \le x_2 < 1 \\

0 & otherwise

\end{array}

\right. $

I can find the marginal pdf of $\displaystyle X_1$ and $\displaystyle X_2$

$\displaystyle \int_{x_2 = x_1}^{x_2=1} 2dx_2 = 2(1 - x_1)$ for $\displaystyle 0 \le x_1 \le 1$

$\displaystyle \int_{x_1 = 0}^{x_1=x_2} 2dx_1 = 2x_2$ for $\displaystyle 0 \le x_2 \le 1$

$\displaystyle X_1$ and $\displaystyle X_2$ are independent.

I can understand up till here. This is where I'm confused. How do you work out the expected values of each random variable from the marginal pdf's of each?

I know: The expected values of $\displaystyle X_1$ and $\displaystyle X_2$ are

$\displaystyle E[X_1] = \frac{1}{3}$

$\displaystyle E[X_2] = \frac{2}{3}$

Can anyone explain the working to get the expected values in more detail?