Observer effect (physics)

2013-12-11T23:49:56Z

70.112.12.124: Removed because this has nothing to do with the "observer effect", but is actually a mathematical certainty due to the uncertainty principle, i.e. having infinitely better instrumentation wouldn't change the result.

{{Unreferenced|date=August 2008}}

In [[probability theory]], the '''multidimensional Chebyshev's inequality''' is a generalization of [[Chebyshev's inequality]], which puts a bound on the probability of the event that a [[random variable]] differs from its [[expected value]] by more than a specified amount.

Let ''X'' be an ''N''-dimensional [[random vector]] with [[expected value]] <math>\mu=\mathbb{E} \left[ X \right] </math> and [[covariance matrix]]

: <math>V=\mathbb{E} \left[ \left(X - \mu \right) \left( X - \mu \right)^T \right]. \, </math>

If <math>V</math> is a [[positive-definite matrix]], for any [[real number]] <math>t>0</math>:
:<math>
\mathrm{Pr}\left( \sqrt{\left( X-\mu\right)^T \, V^{-1} \, \left( X-\mu\right) } > t \right) \le \frac{N}{t^2}
</math>

==Proof==
Since <math>V</math> is positive-definite, so is <math>V^{-1}</math>.
Define the random variable
:<math>
y = \left( X-\mu\right)^T \, V^{-1} \, \left( X-\mu\right) .
</math>
Since <math>y</math> is positive, [[Markov's inequality]] holds:
:<math>
\begin{align}\mathrm{Pr}\left( \sqrt{\left( X-\mu\right)^T \, V^{-1} \, \left( X-\mu\right) } > t\right) &= \mathrm{Pr}\left( \sqrt{y} > t\right)\\
&=\mathrm{Pr}\left( y > t^2 \right) \\
&\le \frac{\mathbb{E}[y]}{t^2} .\end{align}
</math>
Finally,
:<math>\begin{align}\mathbb{E}[y] &= \mathbb{E}[\left( X-\mu\right)^T \, V^{-1} \, \left( X-\mu\right)]\\
&=\mathbb{E}[ \mathrm{trace} ( V^{-1} \, \left( X-\mu\right) \, \left( X-\mu\right)^T )]\\
&= \mathrm{trace} ( V^{-1} V ) = N \end{align}.</math>

[[Category:Probabilistic inequalities]]
[[Category:Statistical inequalities]]

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Observer effect (physics)