Review of Probability Theory

# Review of Probability Theory - Random Processes Te xtbook...

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1 Random Processes Textbook: Probability, Random Variables and Stochastic Processes, 4th edition, By Athanasios Papoulis and S. U. Pillai Teacher: Ø 8 ª Office: 805 Tel: ext. 31822 email: [email protected]

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2 Review of Probability Theory
3 Probability Define a probability space S Y8½ª experimental outcomes i ξ event A i A S set theory È›

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4 Probability Assign to each event A a number ( ) P A probability Axioms 1. ( ) P A 0 2. ( ) 1 P S = 3. If { } AB φ = then ( ) ( ) ( ) P A B P A P B + = + Countable spaces S N outcomes N a finite number ( ) i i P P ξ= è8½ª event probability
5 Probability a S uncountable infinity of elements \8½ ª * outcomes probability event probability \ ª define probability Conditional probability ( ) ( | ) ( ) P AU P A U P U = ( ) 0 P U U i event

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6 Probability Total probability 1 1 ( ) ( | ) ( ) ( | ) ( ) n n P B P B A P A P B A P A = + + event set { } 1 n A A S partition Note: for and i j i i A A i j A S φ ∩ = ≠ = 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 =
7 Probability experiments Cartesian Product 1 2 S S S = × events Cartesian Product A B × 1 2 , S S independent experiments ( ) ( ) ( ) P A B P A P B × = for 1 2 , A S B S Bernoulli trials set n i elements˜ [8½ ª * k i elements subsets ! !( )! n n k k n k   =   -   ˜ [ ª elements˜ [8½ ª permutation

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8 Probability coin with ( ) P h p = coin n i h i i k i ( ) k n k n n P k p q k -   =     1 q p = - a experiment space S event A x ¡J½ ª p i i experiments n i event A k ¡J½ ª*
9 Random Variable (RV) a probability space S experimental outcomes i ξ x R { } x x event Ax ^8½ ª * assign x i outcomes i x^8½ª * define 1 2 { } x x x event a RV assign a number ( ) x to every outcome { } x x event for any x i ( ) 1 and ( ) 0 P x P x < ∞ = = -∞ = i.e., event define subset of S x^8½ ª * define R

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10 Random Variable (RV) RV 8 distribution function ( ) { } F x P x = x x for x -∞ < < ∞ 1 cumulative distribution function (cdf) a countable experiment 1 RV1 cdf 1 staircase function Percentile1 The u percentile of RV x is the smallest number u x i s.t. { } ( ) u u u P x F x = = x
11 Random Variable (RV) cdf L “S 1. ( ) 1 F +∞ = 1 ( ) 0 F -∞ = 2. If 1 2 x x < then 1 2 ( ) ( ) F x F x 3. If 0 ( ) 0 F x = then ( ) 0 F x = for 2200 0 x x 4. { } 1 ( ) P x F x = - x

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Random Variable (RV) 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
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Review of Probability Theory - Random Processes Te xtbook...

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