Alec's sample mean bound
From Maths
Revision as of 01:28, 27 July 2017 by Alec (Talk | contribs) (Created page with "{{Stub page|grade=A*|msg=Needs some work, like what is a random variable for which expectation and variance are defined? Can we have complex or vector ones for example?}} ==In...")
Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Needs some work, like what is a random variable for which expectation and variance are defined? Can we have complex or vector ones for example?
Inequality
Let [ilmath]X_1,\ldots,X_n[/ilmath] be a collection of [ilmath]n[/ilmath] random variables which are pairwise independent, such that:
- [ilmath]\exists\mu\forall i\in\{1,\ldots,n\}\big[\mathbb{E}[X_i]\eq\mu\big][/ilmath] - all of the [ilmath]X_i[/ilmath] have the same expectation and
- Alternatively: [ilmath]\forall i,j\in\{1,\ldots,n\}\big[\mathbb{E}[X_i]\eq\mathbb{E}[X_j]\big][/ilmath], but note we need [ilmath]\mu[/ilmath] in the expression
- [ilmath]\exists\sigma\forall i\in\{1,\ldots,n\}\big[\text{Var}(X_i)\eq\sigma^2\big][/ilmath] - all the [ilmath]X_i[/ilmath] have the same variance
- Alternatively: [ilmath]\forall i,j\in\{1,\ldots,n\}\big[\text{Var}(X_i)\eq\text{Var}(X_j)\big][/ilmath], but note again we need [ilmath]\sigma[/ilmath] in the expression
Then
- For all [ilmath]\epsilon>0[/ilmath] we have:
- [math]\mathbb{P}\left[\left\vert\frac{\sum^n_{i\eq 1}X_i}{n}-\mu\right\vert<\epsilon\right]\ge 1-\frac{\sigma^2}{\epsilon^2n}[/math]
- Note that the notation here differs from that in my 2011 research journal slightly, but [ilmath]\sigma[/ilmath] and [ilmath]\mu[/ilmath] were present.
- [math]\mathbb{P}\left[\left\vert\frac{\sum^n_{i\eq 1}X_i}{n}-\mu\right\vert<\epsilon\right]\ge 1-\frac{\sigma^2}{\epsilon^2n}[/math]
- [math]\mathbb{P}\left[\left\vert\frac{\sum^n_{i\eq 1}X_i}{n}-\mu\right\vert<\epsilon\right]\ge 1-\frac{\sigma^2}{\epsilon^2n}[/math]
