Review of Probability Theory

# Review of Probability Theory - Review of Probability Theory...

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Review of Probability Theory Probability Define a probability space S experimental outcomes i ξ e event A i i ξ Ê A S i i set theory Assign to each event A a number ( ) P A e probability Axiomse 1. ( ) P A 0 e 2. ( ) 1 P S = 3. If { } AB φ = then ( ) ( ) ( ) P A B P A P B + = + Countable spacese S N e outcomese N e a finite number ( ) i i P P ξ = è event e probabilitye S uncountable infinity of elements outcomes e probability e e event e probability define probabilitye Conditional probability ( ) ( | ) ( ) P AU P A U P U = e ( ) 0 P U i i U i evente Total probability 1 1 ( ) ( | ) ( ) ( | ) ( ) n n P B P B A P A P B A P A = + + L event set { } 1 n A A L e S partitione Note: for and i j i i A A i j A S φ = = U Bayes’ Theorem ( | ) ( ) ( | ) ( ) i i i P B A P A P A B P B = Events A and B are independent, if ( ) ( ) ( ) P AB P A P B =

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° experiments e Cartesian Product e 1 2 S S S = events e Cartesian Product A B i 1 2 , S S independent experiments e ( ) ( ) ( ) P A B P A P B i = for e 1 2 , A S B S U Bernoulli trialse set e n i elements k i elements e subsets e e ! !( )! n n k k n k �� = �� - �� x elements permutatione coin with ( ) P h p = Η coin n i h i k i ( ) k n k n n P k p q k - �� = �� �� U 1 q p = - experiment space S e event A p experiments n i i event A e k ± Random Variable (RV) probability space S e experimental outcomes i ξ e x R i { } x i x event A assign e x i outcomes i ξ H define 1 2 { } x x i x e evente RV e assign a number ( ) x ξ to every outcome ξ { } x i x event for any x i ( ) 1 and ( ) 0 P x P x < = = - = i.e., event e define e subset of S define e R e RV distribution function ( ) { } F x P x = x x for x - < < cumulative distribution function (cdf) countable experiment e RVe cdf e staircase function Percentilee The u percentile of RV x is the smallest number u x i s.t.
{ } ( ) u u u P x F x = = x cdf M 1. ( ) 1 F + = e ( ) 0 F - = 2. If 1 2 x x < then 1 2 ( ) ( ) F x F x i 3. If 0 ( ) 0 F x = then ( ) 0 F x = for 2200 0 x x i 4. { } 1 ( ) P x F x = - x 5. ( ) F x is continuous from the right, i.e., ( ) ( ) F x F x + = 6. 1 2 2 1 { } ( ) ( ) P x x F x F x < = - x 7. { } ( ) ( ) P x F x F x - = = - x 8. 1 2 2 1 { } ( ) ( ) P x x F x F x - = - x The density function ( ) ( ) dF x f x dx = e pdfe probability density function ( ) 0 f x i i ( ) 1 f x dx i - = { } 2 1 1 2 ( ) x x P x x x f x dx < = Gaussian function ( 29 2 1 / 2 2 x g x e π - = ( 29 ( 29 2 / 2 1 2 x x y G x g y dy e dy π - - - = = ( 29 ( 29 2 / 2 0 1 1 2 2 x y erf x G x e dy π - = - = ÷ ± M Normal or Gaussian RV 2 2 ( ) / 2 1 1 ( ) ( ) 2 x x f x g e η σ η σ σ σ π - - - = = ( ) ( ) x F x G η σ - =

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Mixture Gaussian Distribution or GMM (Gaussian Mixture Model) 2 1 ( ) ( ; , ) K n k n k k k P x c N x μ σ = Θ = e k e mixture componente 2 ={ , , 1, , } k k k c k K μ σ Θ = L
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