# Covariance formula

by Marco Taboga, PhD

The covariance between two random variables
and
can be computed using the definition of
covariance:where
the capital letter
indicates the expected value operator.

## Formula for discrete variables

When the two random variables are
discrete, the above formula can be
written
aswhere
is the set of all couples of values of
and
that can possibly be observed and
is the probability of observing a specific couple
.
This sum is a weighted average of the products of the deviations of the two
random variables from their respective means.

To see how to apply this formula, read some
Solved exercises.

## Formula for continuous variables

When the two random variables, taken together, form a
continuous random
vector, the formula can be expressed as a double
integral:where
is the joint probability
density function of
and
.

To see how to apply this formula, read some
Solved exercises.

## A simple covariance formula

Using the formulae above to compute covariance can sometimes be tricky. This
is the reason why the following simpler (and equivalent) **covariance
formula** is often
used:

For instance, this formula is straightforward to use when we know the
joint moment generating function of
and
.
Taking partial derivatives of the joint moment generating function, we can
derive the moments
,
and
and then plug their values in this formula.

## More details

In the lecture entitled
Covariance you can find
more details about this formula, including a **proof** of it and
some **exercises**.

## Keep reading the glossary

Previous entry: Countable additivity

Next entry: Covariance stationary