Contents
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}. $
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
