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





Any help in figuring out how to present this stuff, what to name pages, and what to transclude into what, is appreciated.

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

Let there be a Probability Density Function $f_X$ that is riemann integrable with $\displaystyle \int_{\mathop \to -\infty}^{\mathop \to +\infty} \map {f_X} x \, \rd x = 1$

Then the expectation of $X$, written $\expect X$, is defined as:


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

whenever this improper integral exists and either:


 * converges absolutely

or:


 * diverges to $+\infty$ or $-\infty$