Difference between revisions of "Variance"

Latest revision as of 19:45, 24 July 2016

Given an integrable random variable $X$ we define the variance of $X$ as follows:

[Expand]
Theorem:
$\text{Var}(X)=\mathbb{E}[X^2]-(\mathbb{E}[X])^2$

$=\mathbb{E}\left[X^2-2X\mu+\mu^2\right]$

$=\mathbb{E}\left[X^2\right]-2\mu\mathbb{E}[X]+\mu^2$

But!

$\mu=\mathbb{E}[X]$

$=\mathbb{E}\left[X^2\right]-2\mu^2+\mu^2$

$=\mathbb{E}\left[X^2\right]-\mu^2$

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

As required.

@@ Line 1: / Line 1: @@
 ==Definition==
-Given a [[Random variable|random variable]] {{M|X}} we define the '''variance''' of {{M|X}} as follows:
+Given an [[Integral (measure theory)|integrable]] [[Random variable|random variable]] {{M|X}} we define the '''variance''' of {{M|X}} as follows:
 * <math>\text{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right]</math> where {{M|\mu}} is the ''mean'' or [[Expected value|expected value]] of {{M|X}}
 ==Other forms==