Joint Expectation

\[ \mathbb{E}[XY]= \sum_{\Omega_X}\sum_{\Omega_Y}x y \cdot p_{X,Y}(x,y) \text{ or} \\ \mathbb{E}[XY]= \int_{\Omega_X}\int_{\Omega_Y}x y \cdot f_{X,Y}(x,y) \]

• Why is this useful? Because it leads to the correlation and covariance.

  The book spends some time justifying this for the discrete case, but I am ok just taking this as given:

• Covariance of two variables X and Y is:

\[ \begin{align} Cov(X,Y) =& \mathbb{E}[XY]- \mathbb{E}[X]\mathbb{E}[Y] \\ =& \mathbb{E}[(X-\mu_x)(Y- \mu_y)] \end{align} \] where \(\mu_x = \mathbb{E}[X]\) and \(\mu_y = \mathbb{E}[Y]\)

• Covariance allows use to state this theorem:

\[ \mathbb{E}[X+Y] = \mathbb{E}[X] + \mathbb{E}[Y] \\ Var[X+Y] = Var[X] + 2 Cov(X,Y) + Var[Y] \]