While Expectation can be a useful measurement of a r.v., it doesn’t tell us much info about how “spread out” the distribution of all the values of are. If we want a measure of how much tends to deviate from its expectation, we can use the variance.

Variance of a r.v. is given by

Another useful, equivalent representation is

which is used more often in practice, because we just need to calculate the expectation and then use something like LOTUS to get .

Properties of variance

and in fact, unless is a constant (its support size is 1). This is a good sanity check to see if you’re on the right path.

for any constant . This makes sense: we are shifting every support by a constant amount. This will probably affect , but the variance remains the same.

for any constant . For example, if we want to convert from hours to minutes, we have to multiple variance by , but we only multiply expectation and standard deviation by .

Variance of the sum of two variables

The general formula is always true assuming that the expectation is well defined:


Two more results follow immediately from the general formula:

  • If and are Independent, then . However, the equation being true does not imply independence.

Standard deviation

For this is given by

In most applications, it is easier to interpret the standard deviation than the variance of a random variable.