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Often, we are interested in two random variables $X$ and $Y$, and want to understand how they are related. For instance, $X$ might be tomorrow's high temperature, and $Y$ might be tomorrow's barometric pressure. Or $X$ might be the concentration of virus in a patient's blood, and $Y$ might be the concentration of an antibody. Or $X$ and $Y$ might be the horizontal and vertical coordinates of the spot where a dart hits a board. When we have two random variables $X$ and $Y$, the pair $(x,y)$ takes values in the plane, and we speak of the probability per unit area of $(x,y)$ landing in a certain region. This is called the joint probability density function, and is written $f_{X,Y}(x,y)$.
If we know the joint distribution of $X$ and $Y$, we can compute the distribution of just $X$ or just $Y$. These are called marginal distributions, because for discrete random variables they are often written on the margins of a table. We get $f_X(x)$ by integrating $f_{X,Y}(x,y)$ over $y$, and $f_Y(y)$ by integrating $f_{X,Y}(x,y)$ over $x$: \begin{eqnarray*} f_X(x) & = & \int_{\infty}^\infty f_{X,Y}(x,y) dy \\ f_Y(y) & = & \int_{\infty}^\infty f_{X,Y}(x,y) dx \end{eqnarray*} If $\mu_X$ and $\mu_Y$ are the average values of $X$ and $Y$, then the covariance of $X$ and $Y$ is $$Cov(X,Y) = E((X\mu_X)(Y\mu_Y)) = \iint (x\mu_x)(y\mu_y) f_{X,Y}(x,y) dA.$$ Just like the variance of one variable, this is more easily computed as $$Cov(X,Y) = E(XY)  \mu_X\mu_Y = \iint xy f_{X,Y}(x,y) dA  \mu_X\mu_Y.$$ The correlation between $X$ and $Y$ is $$Cor(X,Y) = \frac{Cov(X,Y)}{\sigma_X\sigma_Y}.$$ This is a number, often written $r$, between $1$ and $1$. If $r$ is close to 1, then all the likely values of $(X,Y)$ lie close to a straight line of positive slope. In that case we say that $X$ and $Y$ are positively correlated. Increases in $X$ are then associated with increases in $Y$, and viceversa. This is statistical evidence that $X$ and $Y$ are somehow related. Maybe $X$ causes $Y$, or maybe $Y$ causes $X$, or maybe something else causes both of them. If $r$ is close to $1$, then the likely values of $(X,Y)$ lie close to a line of negative slope, so increasing $X$ tends to decrease $Y$, and viceversa. $X$ and $Y$ are said to be negatively correlated. Again, this is evidence that $X$ and $Y$ are related. If $r$ is close to 0, then $X$ and $Y$ are said to be uncorrelated.
