Difference between revisions of "Cheat3 ECE302Fall2008sanghavi" - Rhea

Revision as of 17:14, 18 November 2008

Covariance

• $ COV(X,Y)=E[(X-E[X])(Y-E[Y])]\! $
• $ COV(X,Y)=E[XY]-E[X]E[Y]\! $

Correlation Coefficient

$ \rho(X,Y)= \frac {cov(X,Y)}{\sqrt{var(X)} \sqrt{var(Y)}} \, $

Markov Inequality

Loosely speaking: In a nonnegative RV has a small mean, then the probability that it takes a large value must also be small.

• $ P(X \geq a) \leq E[X]/a\! $

for all a > 0

Chebyshev Inequality

"Any RV is likely to be close to its mean"

$ \Pr(\left|X-E[X]\right|\geq C)\leq\frac{var(X)}{C^2}. $

Weak Law of Large Numbers

The weak law of large numbers states that the sample average converges in probability towards the expected value

$ \overline{X}_n \, \xrightarrow{P} \, \mu \qquad\textrm{for}\qquad n \to \infty. $

ML Estimation Rule

$ \hat a_{ML} = \text{max}_a ( f_{X}(x_i;a)) $ continuous

$ \hat a_{ML} = \text{max}_a ( Pr(x_i;a)) $ discrete

MAP Estimation Rule

$ \hat \theta_{MAP} = \text{argmax}_\theta ( f_{\theta|X}(\theta|x)) $

Which can be expanded and turned into the following (if I am not mistaken):

$ \hat \theta_{MAP} = \text{argmax}_\theta ( f_{X|\theta}(x|\theta)f_{\theta}(\theta)) $

Bias of an Estimator, and Unbiased estimators

An estimator is unbiased if: $ E[\hat a_{ML}] = a $ for all values of a

Confidence Intervals, and how to get them via Chebyshev

• $ \theta \text{ is unknown and fixed} * \hat \theta \text{ is random and should be close to} \theta \text{ most of the time} * if Pr[|\hat \theta - \theta|] <= \text {(1-a) then we say we have (1-a) confidence in the interval} [\hat \theta - E, \hat \theta + E] $