Editing Variance (section)

==Definition==
The variance of a random variable <math>X</math> is the [[expected value]] of the [[Squared deviations from the mean|squared deviation from the mean]] of <math>X</math>, <math>\mu = \operatorname{E}[X]</math>:
<math display="block"> \operatorname{Var}(X) = \operatorname{E}\left[(X - \mu)^2 \right]. </math>
This definition encompasses random variables that are generated by processes that are [[discrete random variable|discrete]], [[continuous random variable|continuous]], [[Cantor distribution|neither]], or mixed. The variance can also be thought of as the [[covariance]] of a random variable with itself:

<math display="block">\operatorname{Var}(X) = \operatorname{Cov}(X, X).</math> 
The variance is also equivalent to the second [[cumulant]] of a probability distribution that generates <math>X</math>. The variance is typically designated as <math>\operatorname{Var}(X)</math>, or sometimes as <math>V(X)</math> or <math>\mathbb{V}(X)</math>, or symbolically as <math>\sigma^2_X</math> or simply <math>\sigma^2</math> (pronounced "[[sigma]] squared"). The expression for the variance can be expanded as follows:
<math display="block">\begin{align}
\operatorname{Var}(X) &= \operatorname{E}\left[{\left(X - \operatorname{E}[X]\right)}^2\right] \\[4pt]
&= \operatorname{E}\left[X^2 - 2 X \operatorname{E}[X] + \operatorname{E}[X]^2\right] \\[4pt]
&= \operatorname{E}\left[X^2\right] - 2 \operatorname{E}[X] \operatorname{E}[X] + \operatorname{E}[X]^2 \\[4pt]
&= \operatorname{E}\left[X^2\right] - 2\operatorname{E}[X]^2 + \operatorname{E}[X]^2 \\[4pt]
&= \operatorname{E}\left[X^2\right] -  \operatorname{E}[X]^2
\end{align}</math>

In other words, the variance of {{mvar|X}} is equal to the mean of the square of {{mvar|X}} minus the square of the mean of {{mvar|X}}.  This equation should not be used for computations using [[floating-point arithmetic]], because it suffers from [[catastrophic cancellation]] if the two components of the equation are similar in magnitude. For other numerically stable alternatives, see [[algorithms for calculating variance]].

===Discrete random variable===

If the generator of random variable <math>X</math> is [[Discrete probability distribution|discrete]] with [[probability mass function]] <math>x_1 \mapsto p_1, x_2 \mapsto p_2, \ldots, x_n \mapsto p_n</math>, then

<math display="block">\operatorname{Var}(X) = \sum_{i=1}^n p_i \cdot {\left(x_i - \mu\right)}^2,</math>

where <math>\mu</math> is the expected value. That is,

<math display="block">\mu = \sum_{i=1}^n p_i x_i .</math>

(When such a discrete [[weighted variance]] is specified by weights whose sum is not&nbsp;1, then one divides by the sum of the weights.)

The variance of a collection of <math>n</math> equally likely values can be written as

<math display="block"> \operatorname{Var}(X) = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2 </math>

where <math>\mu</math> is the average value. That is,

<math display="block">\mu = \frac{1}{n}\sum_{i=1}^n x_i .</math>

The variance of a set of <math>n</math> equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other:<ref>{{cite conference|author=Yuli Zhang |author2=Huaiyu Wu |author3=Lei Cheng |date=June 2012|title=Some new deformation formulas about variance and covariance|conference=Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012)|pages=987–992}}</ref>

<math display="block"> \operatorname{Var}(X) = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \frac{1}{2} {\left(x_i - x_j\right)}^2 = \frac{1}{n^2} \sum_i \sum_{j>i} {\left(x_i - x_j\right)}^2. </math>

===Absolutely continuous random variable===

If the random variable <math>X</math> has a [[probability density function]] <math>f(x)</math>, and <math>F(x)</math> is the corresponding [[cumulative distribution function]], then

<math display="block">\begin{align}
   \operatorname{Var}(X) = \sigma^2 &= \int_{\R} {\left(x - \mu\right)}^2 f(x) \, dx \\[4pt]
     &= \int_{\R} x^2 f(x)\,dx -2\mu\int_{\R} xf(x)\,dx + \mu^2\int_{\R} f(x)\,dx \\[4pt]
     &= \int_{\R} x^2 \,dF(x) - 2 \mu \int_{\R} x \,dF(x) + \mu^2 \int_{\R} \,dF(x) \\[4pt]
     &= \int_{\R} x^2 \,dF(x) - 2 \mu \cdot \mu + \mu^2 \cdot 1 \\[4pt]
     &= \int_{\R} x^2 \,dF(x) - \mu^2,
 \end{align}</math>

or equivalently,

<math display="block">\operatorname{Var}(X) = \int_{\R} x^2 f(x) \,dx - \mu^2 ,</math>

where <math>\mu</math> is the expected value of <math>X</math> given by

<math display="block">\mu = \int_{\R} x f(x) \, dx = \int_{\R} x \, dF(x). </math>

In these formulas, the integrals with respect to <math>dx</math> and <math>dF(x)</math>
are [[Lebesgue integral|Lebesgue]] and [[Lebesgue–Stieltjes integration|Lebesgue–Stieltjes]] integrals, respectively.

If the function <math>x^2f(x)</math> is [[Riemann-integrable]] on every finite interval <math>[a,b]\subset\R,</math> then

<math display="block">\operatorname{Var}(X) = \int^{+\infty}_{-\infty} x^2 f(x) \, dx - \mu^2, </math>

where the integral is an [[improper Riemann integral]].