Probability

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Sample Space

  • random variable: is a function from sample space into the real numbers
  • $\sigma-algebra$
  • Power set

Permutations and Combinations

  • Vandermonde Identity

Binomial and Multinomial Distribution

  • connection to hypergeometric distribution: binomial is with replacement, and hypergeometric is without replacement.

Bayes Rule

  • Prior

  • Posterior

  • Conditional Independence

Events $A$ and $B$ are conditional independence give $C$ if

\[P(A\cap{B}\|C)=P(A\|C)P(B\|C)\]

Example: Chess Competition, if you play chess with an opponent. It may happen that the games are conditional independent but not independent.

  • Short form
  • w/o without
  • w/ with
  • b/c because

CDF, PDF, PMF

  • Cumulative Density Function
  • Probability Density Function
  • Probability Mass Function

Maximum Likelihood Estimation