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

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.

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.

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.

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

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