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! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Cauchy-schwarz Inequality | ! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Cauchy-schwarz Inequality | ||
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| − | + | | <math> \vert a_1b_1 + a_2b_2 + \cdots + a_nb_n \vert ^2 \leqq \left ( \vert a_1 \vert ^2 + \vert a_2 \vert ^2 + \cdots + \vert a_n \vert ^2 \right ) \left ( \vert b_1 \vert ^2 + \vert b_2 \vert ^2 + \cdots + \vert b_n \vert ^2 \right )</math> | |
| − | | <math>\ | + | |
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| align="right" style="padding-right: 1em;" | Gaussian random variable with parameter <math>\mu \mbox{ and } \sigma^2</math> | | align="right" style="padding-right: 1em;" | Gaussian random variable with parameter <math>\mu \mbox{ and } \sigma^2</math> | ||
Revision as of 07:02, 25 November 2010
| Inequalities | |
|---|---|
| Triangular Inequalities | |
| $ \vert a_1 \vert - \vert a_2 \vert \leqq \vert a_1 +a_2 \vert \leqq \vert a_1 \vert + \vert a_2 \vert $ | |
| $ \vert a_1 + a_2 + \cdots + a_n \vert \leqq \vert a_1 \vert + \vert a_2 \vert + \cdots + \vert a_n \vert $ | |
| Cauchy-schwarz Inequality | |
| $ \vert a_1b_1 + a_2b_2 + \cdots + a_nb_n \vert ^2 \leqq \left ( \vert a_1 \vert ^2 + \vert a_2 \vert ^2 + \cdots + \vert a_n \vert ^2 \right ) \left ( \vert b_1 \vert ^2 + \vert b_2 \vert ^2 + \cdots + \vert b_n \vert ^2 \right ) $ | |
| Gaussian random variable with parameter $ \mu \mbox{ and } \sigma^2 $ | $ \,E[X] = \mu,\ \ Var(X) = \sigma^2\, $ |
| Exponential random variable with parameter $ \lambda $ | $ \,E[X] = \frac{1}{\lambda},\ \ Var(X) = \frac{1}{\lambda^2}\, $ |
| Inequalities Involving Arithmetic, Geometric and Harmonic | |
| Holder Inequality | |
| Tchebytchev Inequality | |
| Minkowski Inequality | |
| Cauchy-schwarz Inequality for Integrals | |
| Holder Inequality for Integrals | |
| Minkowski Inequality for Integrals | |
