The covariance of $X$ and $Y$ is the number defined by \[Cov(X,Y) = E((X-\mu_X)(Y-\mu_Y)).\]
The correlation of $X$ and $Y$ is the number defined by \[\rho_{XY}=\frac{Cov(X,Y)}{\sigma_X \sigma_Y}.\] The value $\rho_{XY}$ is also called the correlation coefficient
Theorem 4.5.1 For any random variables $X$ and $Y,$ \[Cov(X,Y) = EXY - \mu_X \mu_Y.\]
Theorem 4.5.2 If $X$ and $Y$ are independent random variables, then $Cov(X,Y)=0$ and $\rho_{XY}=0.$
Theorem 4.5.3 If $X$ and $Y$ are any two random variables and $a$ and $b$ are any two constants, then \[Var(aX + bY) = a^2VarX + b^2VarY + 2abCov(X,Y).\] If $X$ and $Y$ are independent random variables, then \[Var(aX + bY) = a^2VarX + b^2VarY\]
Theorem 4.5.4
For any random variables $X$ and $Y,$
a) $-1\leq \rho_{XY} \leq 1.$
b) $|\rho_{XY}| = 1$ if and only if there exist numbers $a\neq 0$ and $b$ such that $P(Y=aX +b) = 1.$ If $\rho_{XY}=1,$ then $a>0,$ and if $\rho_{XY} = -1,$ then $a<0.$
Let $-\infty < \mu_X < \infty, -\infty < \mu_Y < \infty, \;0 < \sigma_X,\; 0< \sigma_Y,$ and $-1 < \rho < 1$ be five real numbers. The bivariate normal pdf with means $\mu_X$ and $\mu_Y,$ variances $\sigma_X^2$ and $\sigma_Y^2,$ and correlation $\rho$ is the bivariate pdf given by \[f(x,y) = \left(2\pi\sigma_x \sigma_Y \sqrt{1-\rho^2}\right)^{-1}\exp\left(-\frac{1}{2(1-\rho^2)}\left(\left(\frac{x-\mu_X}{\sigma_X}\right)^2-2\rho\left(\frac{x-\mu_X}{\sigma_X}\right)\left(\frac{y-\mu_Y}{\sigma_Y}\right)+\left(\frac{y-\mu_Y}{\sigma_Y}\right)^2\right)\right)\] for $-\infty < x < \infty$ and $-\infty < y < \infty.$
Properties of this distribution include:
a. The marginal distribution of $X$ is $N(\mu_X, \sigma_X^2).$
b. The marginal distribution of $Y$ is $N(\mu_Y, \sigma_Y^2).$
c. The correlation between $X$ and $Y$ is $\rho_{XY} = \rho.$
d. For any constants $a$ and $b,$ the distribution of $aX+bY$ is $N(a\mu_X + b\mu_Y, a^2\sigma_X^2 + b^2\sigma_Y^2 + 2ab\rho\sigma_X\sigma_Y).$