variance

Definition

The variance of a real-valued random variable $X$ is

$$\mathrm{Var}X=𝔼\left[{(X-m)}^2\right],m=𝔼X,$$

provided that both expectations $𝔼X$ and $𝔼[{(X-m)}^2]$ exist.

The variance of $X$ is often denoted by ${\sigma }^2(X)$, ${\sigma }_X^2$,or simply ${\sigma }^2$.The exponent on $\sigma$ is put there so that the number$\sigma =\sqrt{{\sigma }^2}$is measured in the same units as the random variable $X$itself.

The quantity $\sigma =\sqrt{\mathrm{Var}X}$ is called the standard deviationof $X$;because of the compatibility of the measuring units,standard deviation is usually the quantity that is quotedto describe an emprical probability distribution, rather than the variance.

Usage

The variance is a measure of the dispersion or variationof a random variableabout its mean $m$.

It is not always the best measure of dispersion for all random variables,but compared to other measures,such as the absolute mean deviation, $𝔼[|X-m|]$,the variance is the most tractable analytically.

The variance is closely related to the ${𝐋}^2$ norm forrandom variables over a probability space.

Properties

1. The variance of $X$ is the second moment of $X$ minusthe square of the first moment:

   $$\mathrm{Var}X=𝔼[X^2]-𝔼{[X]}^2.$$

   This formula is often used to calculate variance analytically.

2. Variance is not a linear function. It scales quadratically,and is not affected by shifts in the mean of the distribution:

   $$\mathrm{Var}[aX+b]=a^2\mathrm{Var}X,\text{ for any }a,b\in ℝ\text{.}$$

3. A random variable $X$ is constant almost surely if and onlyif $\mathrm{Var}X=0$.

4. The variance can also be characterized asthe minimum of expected squared deviation of a random variable from any point:

   $$\mathrm{Var}X=\underset{a\in ℝ}{inf}𝔼[{(X-a)}^2].$$

5. For any two random variables $X$ and $Y$ whose variances exist,the variance of the linear combination $aX+bY$can be expressed in terms of their covariance:

   $$\mathrm{Var}[aX+bY]=a^2\mathrm{Var}X+b^2\mathrm{Var}Y+2ab\mathrm{Cov}[X,Y],$$

   where $\mathrm{Cov}[X,Y]=𝔼[(X-𝔼X)(Y-𝔼Y)]$,and $a,b\in ℝ$.

6. For a random variable $X$, with actual observations $x_1,\mathrm{\dots },x_n$,its variance is often estimatedempirically with the sample variance:

   $$\mathrm{Var}X\approx s^2=\frac{1}{n-1}\sum _{i=1}^n{(x_i-\overline{x})}^2,\overline{x}=\frac{1}{n}\sum _{j=1}^n{x}_j.$$