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.  In the textbook, what is the probability of the die being a one if you know that the face number is odd? (1/3).  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:  P ( A | B ) 0 for any event A .  P ( A | B ) = 1 for any event A B.  If { A i B }, i =1,2,3, are mutually exclusive, then  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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