Variance and Standard Deviation

Variance

Definition: Variance

The variance of a random variable $X$ is the Expectation of square of the deviation from $X$‘s Expectation:

$$\mathbb{E}[(X-\mathbb{E}[X]{)}^2]$$

NOTATION

$$\mathrm{Var}(X){\sigma }^2{s}^{2}V(X)\mathbb{V}(X)$$

Tip: Discrete Variance

The variance of a discrete random variable $X$ with support $S=\{x_1,x_2,\dots \}$ and probability mass function $p$ is given by the value of the following series:

$${\sigma }^2=\sum _{i=1}^{\infty }p(x_i)\cdot (x_i-\mathbb{E}[X]{)}^2$$

If $S$ is finite, then this reduces to a simple sum:

$${\sigma }^2=\sum _{i=1}^{|S|}p(x_i)\cdot (x_i-\mathbb{E}[X]{)}^2$$

Properties

Theorem: Computation Formula for Variance

The variance of every random variable $X$ is the difference between the Expectation of its square and the square of its Expectation:

$${\sigma }^2=\mathbb{E}[X^2]-\mathbb{E}[X{]}^2$$

PROOF

TODO

Theorem: Variance of a Constant

The variance of a constant random variable is zero.

PROOF

TODO

Theorem: Variance of Constant Multiple

The variance of a constant multiple $\lambda \in \mathbb{R}$ of a random variable $X$ is equal to the square of $\lambda$ multiplied by the variance of $X$:

$$\mathrm{Var}(\lambda \cdot X)={\lambda }^2\cdot \mathrm{Var}(X)$$

PROOF

TODO

Theorem: Variance of Sum and Difference

If $X$ and $Y$ are two Random Variables, then the variance of their and the variance of their difference are both equal to the sum of their variances:

$$\begin{array}{r}\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)\\ \mathrm{Var}(X-Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)\end{array}$$

PROOF

TODO

Standard Deviation

Definition: Standard Deviation of a Random Variable

The standard deviation of a random variable $X$ is the square-root of its variance.

$$\sqrt{\mathrm{Var}(X)}$$

NOTATION

$$\sigma {\sigma }_{X}s$$