Why are we doing 3 integrals? I thought that when we calculate expected value, it is just E(X) or in this case E(X+Y) and that when we calculate variance, we are just doing $E(X^2)-[E(X)]^2$ or in this case Var(X+Y) = Var(X) + 2Cov(X,Y) + Var(Y)? I am honestly not understanding a lot of concepts...
So for 2 random variables to be "uncorrelated," their covariance has to be 0? (E(XY)-E(X)E(Y))? I wasn't really sure how to start with this problem. The problem says their means are equal but that the variance of X is larger than Y but I am not sure what to do with the equation for W that they...
Suppose that X, Y are uncorrelated random variables which are each measurements of some unknown quantity $\mu$. Both random variables have $\mu_{X} = \mu_{Y} = \mu$, but $\sigma^2_{X} > \sigma^2_{Y}$. Determine the value of $\alpha$ in [0, 1] which will minimize the variance of the random...
A machine consists of two components whose lifetimes have a joint density function:
$f(x,y) = \left\{
\begin{array}{ll}
\frac{1}{50} & \quad x \geq 0, y \geq 0, x+y \leq 10 \\
0 & \quad Otherwise
\end{array}
\right.$
The machine operates until both...