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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

\[ \mathop{\operatorname{Var}} (X) \qquad \sigma^2 \qquad s^2 \qquad V(X) \qquad \mathbb{V}(X) \]

Tip: Discrete Variance

The variance of a discrete random variable \(X\) with support \(S = \{x_1, x_2, \dotsc\}\) 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 \(X\):

\[ \mathop{\operatorname{Var}}(\lambda \cdot X) = \lambda^2 \cdot \mathop{\mathrm{Var}}(X) \]
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:

\[ \begin{aligned} \mathop{\operatorname{Var}}(X + Y) = \mathop{\operatorname{Var}}(X) + \mathop{\operatorname{Var}}(Y) \\ \mathop{\operatorname{Var}}(X - Y) = \mathop{\operatorname{Var}}(X) + \mathop{\operatorname{Var}}(Y) \end{aligned} \]
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.

\[ \sqrt{\mathop{\operatorname{Var}}(X)} \]

Notation

\[ \sigma \qquad \sigma_X \qquad s \]