Difference between revisions of "Variance"
From Maths
(Created page with "==Definition== Given a random variable {{M|X}} we define the '''variance''' of {{M|X}} as follows: * <math>\text{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right]</...") |
(→Definition: integrable r.v.) |
||
| (2 intermediate revisions by one other user not shown) | |||
| Line 1: | Line 1: | ||
==Definition== | ==Definition== | ||
| − | Given | + | 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}} | * <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== | ==Other forms== | ||
{{Begin Theorem}} | {{Begin Theorem}} | ||
| − | Theorem: <math>\text{Var}(X)=\mathbb{E}[X^2] | + | Theorem: <math>\text{Var}(X)=\mathbb{E}[X^2]-(\mathbb{E}[X])^2</math> |
{{Begin Proof}} | {{Begin Proof}} | ||
* <math>\text{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right]</math> | * <math>\text{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right]</math> | ||
| Line 13: | Line 12: | ||
:: But! <math>\mu=\mathbb{E}[X]</math> | :: But! <math>\mu=\mathbb{E}[X]</math> | ||
: <math>=\mathbb{E}\left[X^2\right]-2\mu^2+\mu^2</math> | : <math>=\mathbb{E}\left[X^2\right]-2\mu^2+\mu^2</math> | ||
| − | : <math>=\mathbb{E}\left[X^2\right]-\mu</math> | + | : <math>=\mathbb{E}\left[X^2\right]-\mu^2</math> |
| − | : <math>=\mathbb{E}\left[X^2\right]-\mathbb{E}[X]</math> | + | : <math>=\mathbb{E}\left[X^2\right]-(\mathbb{E}[X])^2</math> |
As required. | As required. | ||
Latest revision as of 19:45, 24 July 2016
Definition
Given an integrable random variable [ilmath]X[/ilmath] we define the variance of [ilmath]X[/ilmath] as follows:
- [math]\text{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right][/math] where [ilmath]\mu[/ilmath] is the mean or expected value of [ilmath]X[/ilmath]
Other forms
Theorem: [math]\text{Var}(X)=\mathbb{E}[X^2]-(\mathbb{E}[X])^2[/math]
- [math]\text{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right][/math]
- [math]=\mathbb{E}\left[X^2-2X\mu+\mu^2\right][/math]
- [math]=\mathbb{E}\left[X^2\right]-2\mu\mathbb{E}[X]+\mu^2[/math]
- But! [math]\mu=\mathbb{E}[X][/math]
- [math]=\mathbb{E}\left[X^2\right]-2\mu^2+\mu^2[/math]
- [math]=\mathbb{E}\left[X^2\right]-\mu^2[/math]
- [math]=\mathbb{E}\left[X^2\right]-(\mathbb{E}[X])^2[/math]
As required.
