Variance

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 $X$ are. If we want a measure of how much $X$ tends to deviate from its expectation, we can use the variance.

Variance of a r.v. $X$ is given by

$$\text{Var}(X)=\mathbb{E}\left((X-\mathbb{E}(X){)}^2\right)$$

Another useful, equivalent representation is

$$\text{Var}(X)=\mathbb{E}(X^2)-\mathbb{E}(X{)}^2$$

which is used more often in practice, because we just need to calculate the expectation and then use something like LOTUS to get $\mathbb{E}(X^2)$.

Properties of variance

$\text{Var}(X)\ge 0$ and in fact, $\text{Var}(X)>0$ unless $X$ is a constant (its support size is 1). This is a good sanity check to see if you’re on the right path.

$\text{Var}(X+c)=\text{Var}(X)$ for any constant $c$. This makes sense: we are shifting every support by a constant amount. This will probably affect $\mathbb{E}(X)$, but the variance remains the same.

$\text{Var}(cX)=c^2\cdot \text{Var}(X)$ for any constant $c$. For example, if we want to convert $X$ from hours to minutes, we have to multiple variance by $3600$, but we only multiply expectation and standard deviation by $60$.

Variance of the sum of two variables

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

$$\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)+2\text{Cov}(X,Y)$$

Corollaries

Two more results follow immediately from the general formula:

• $\text{Var}(X-Y)=\text{Var}(X)+\text{Var}(Y)-2\text{Cov}(X,Y)$
• If $X$ and $Y$ are Independent, then $\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)$. However, the equation being true does not imply independence.

Standard deviation

For $X$ this is given by

$$\text{SD}(X)=\sqrt{\text{Var}(X)}$$

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