User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Symbols:LaTeX Commands/ProofWiki Specific





Characterization of Expectation
Let $X$ be a continuous random variable over the probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $F_X = \map \Pr {X < x}$ be the cumulative probability function of $X$.

Let the Probability Density Function of $f_X$ satisfy $\dfrac {\d x} {\d y}$.

Then the expectation of $X$ is written $\expect X$, satisfies: probability measure


 * $\expect X = \displaystyle \int_{\mathop \to -\infty}^{\mathop \to +\infty} x \map {f_X} x \, \rd x$

whenever this improper integral exists.

Proof
As $X$ is continuous,

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Radon-Nikodym theorem,

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 * $\displaystyle \int_{\Omega} x \, \rd \!\Pr = \int_{\mathop \to -\infty}^{\mathop \to +\infty} x f_X(x) \, \rd x$

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