Revision as of 17:38, 18 November 2008 by Dvanclev (Talk)

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

Bias of an Estimator, and Unbiased estimators

Unbiased if: $ E[a_ML] = a $

Confidence Intervals, and how to get them via Chebyshev

Alumni Liaison

To all math majors: "Mathematics is a wonderfully rich subject."

Dr. Paul Garrett