Multiple random variables

A lot of the time, we will end up dealing with more than one random variable. When we do have two or more variables, we can inspect the linear relationships between the random variables. We call this the covariance.

If we have two random variables, X and Y, then the covariance is defined as follows:

The following are some of the axioms for the covariance:

• If c is a constant, then _.
• _.
• _.
• _.
• _.
• _.

• , given that X and Y are independent (but it does not imply that the two are independent).
• 

However, sometimes, the covariance doesn't give us the full picture of the correlation between two variables. This could be a result of the variance of X and Y. For this reason, we normalize the covariance as follows and get the correlation:

The resulting value will always lie in the [-1, 1] interval.

This leads us to the concept of conditional distributions, where we have two random variables, X and Y, that are not independent and we have the joint distribution, , from which we can get the probabilities,  and . Then, our distribution is defined as follows:

From this definition, we can find our conditional distribution of X given Y to be as follows:

We may also want to find the conditional expectation of X given Y, which is as follows:

Now, if our random variables are independent, then, , which we know to be true because Y has no effect on X.