View source for Alec's sample mean bound

{{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?}}{{ProbMacros}}
==Notice==
It appears that this is actually 3 inequalities in one, which we shall name as follows:
# [[Alec's remaining probability bound]] - that for {{M|X}} a real and non-negative random variable, {{MM|\forall \alpha\in\mathbb{R}_{>0}\left[\P{X\ge \alpha}\le\frac{\E{X} }{\alpha}\right]}}
# [[Alec's deviation probability bound]] - that for a real random variable (possibly negative) {{M|X}} that {{MM|\forall\beta\in\mathbb{R}_{>0}\left[\mathbb{P}\Big[\big\vert X-\E{X}\big\vert\ge\beta\Big]\le\frac{\Var{X} }{\beta^2} \right]}}
# [[Alec's sample mean bound]] (this page)
==Inequality==
Let {{M|X_1,\ldots,X_n}} be a collection of {{M|n}} {{link|random variable||s}} which are [[pairwise independent]], such that:
* {{M|\exists\mu\forall i\in\{1,\ldots,n\}\big[\mathbb{E}[X_i]\eq\mu\big]}} - all of the {{M|X_i}} have the same [[expectation]] and
** Alternatively: {{M|\forall i,j\in\{1,\ldots,n\}\big[\mathbb{E}[X_i]\eq\mathbb{E}[X_j]\big]}}, but note we need {{M|\mu}} in the expression
* {{M|\exists\sigma\forall i\in\{1,\ldots,n\}\big[\text{Var}(X_i)\eq\sigma^2\big]}} - all the {{M|X_i}} have the same [[variance]] 
** Alternatively: {{M|\forall i,j\in\{1,\ldots,n\}\big[\text{Var}(X_i)\eq\text{Var}(X_j)\big]}}, but note again we need {{M|\sigma}} in the expression
Then
* For all {{M|\epsilon>0}} we have:
** <span style="font-size:1.75em;">{{MM|1=\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} }}</span>
*** Note that the notation here differs from that in my 2011 research journal slightly, but {{M|\sigma}} and {{M|\mu}} were present.
**{{MM|1=\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} }}
==History of form==
When I "discovered" this inequality I was looking to say something like "the chance of the sample mean being within so much of the mean is at least ..." 

I didn't know how to handle {{M|\vert X-\E{X}\vert}} (what we'd now call [[Mdm|{{M|\Mdm{X} }}]]) which is why I applied it to variance, and of course {{M|\big(X-\E{X}\big)^2\ge 0}} - the only condition required for the first inequality.