Lecture 04-2007

# Lecture 04-2007 - LECTURE IV Random Variables and...

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Unformatted text preview: LECTURE IV Random Variables and Probability Distributions I Conditional Probability and Independence In order to define the concept of a conditional probability it is necessary to discuss joint probabilities and marginal probabilities. A joint probability is the probability of two random events. For example, consider drawing two cards from the deck of cards. There are 52x51=2,652 different combinations of the first two cards from the deck. The marginal probability is overall probability of a single event or the probability of drawing a given card. The conditional probability of an event is the probability of that event given that some other event has occurred. &#2; In the textbook, what is the probability of the die being a one if you know that the face number is odd? (1/3). &#2; However, note that if you know that the role of the die is a one, that the probability of the role being odd is 1. Axioms of Conditional Probability: &#2; P ( A | B ) 0 for any event ≥ A . &#2; P ( A | B ) = 1 for any event A ⊃ B. &#2; If { A i ∩ B }, i =1,2,3,… are mutually exclusive, then &#2; If B ⊃ H , B ⊃ G and P ( G ) ≠ 0 then ( 29 ( 29 ( 29 1 2 1 2 P A A P A B P A B = + + K K ( 29 ( 29 ( 29 ( 29 P H B P H P G P G B = Theorem 2.4.1: for any pair of events A and B such that P ( B ) ≠ 0. Theorem 2.4.2 (Bayes Theorem): Let Events A 1 , A 2 , … A n be mutually exclusive such that P ( A 1 ∪ A 2 ∪ … A n )=1 and P ( A i )>0 for each i . Let E be an arbitrary event such that P ( E )>0 . Then ( 29 ( 29 ( 29 P A B P A B P B A = ∑ = = n j j j i i i A P A E P A P A E P E A P 1 ) ( ) | ( ) ( ) | ( ) | ( Another manifestation of this theorem is from the joint distribution function: The bottom equality reduces the marginal probability of event...
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Lecture 04-2007 - LECTURE IV Random Variables and...

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