Difference between revisions of "Variance"

Revision as of 13:23, 24 July 2016

Given a 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 6: / Line 6: @@
 ==Other forms==
 {{Begin Theorem}}
-Theorem: <math>\text{Var}(X)=\mathbb{E}[X^2]+(\mathbb{E}[X])^2</math>
+Theorem: <math>\text{Var}(X)=\mathbb{E}[X^2]-(\mathbb{E}[X])^2</math>
 {{Begin Proof}}
 * <math>\text{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right]</math>