Lecture 04-2007 - Random Variables and Probability...

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Random Variables and Probability Distributions: Lecture IV I. Conditional Probability and Independence A. In order to define the concept of a conditional probability it is necessary to discuss joint probabilities and marginal probabilities. 1. 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. 2. The marginal probability is overall probability of a single event or the probability of drawing a given card. 3. The conditional probability of an event is the probability of that event given that some other event has occurred. a) In the textbook, what is the probability of the die being a one if you know that the face number is odd? (1/3). b) 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. B. Axioms of Conditional Probability: 1.   0 P A B for any event A . 2.   1 P A B for any event AB . 3. If   , 1,2, i A B i  are mutually exclusive, then       1 2 1 2 P A A P A B P A B  4. If , B H B G  and   0 PG then         P H B PH P G B C. Theorem 2.4.1:       P A B P A B PB for any pair of events A and B such that   0 . D. Theorem 2.4.2 (Bayes Theorem): Let Events 12 ,, n A A A be mutually exclusive such that   1 n P A A A and   0 i PA for each i . Let E be an arbitrary event such that   0 PE . Then
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AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture IV Professor Charles B. Moss Fall 2007 2 n j j j i i i A P A E P A P A E P E A P 1 ) ( ) | ( ) ( ) | ( ) | ( 1. Another manifestation of this theorem is from the joint distribution function: ) ( ) | ( ) ( ) , ( i i i i A P A E P A E P A E P 2. The bottom equality reduces the marginal probability of event E n i i i A P A E P E P 1 ) ( ) | ( ) ( 3. This yields a friendlier version of Bayes theorem based on the ratio between the joint and marginal distribution function: ) ( ) , ( ) | ( E P A E P E A P i i E.
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This note was uploaded on 07/18/2011 for the course AEB 6933 taught by Professor Carriker during the Fall '09 term at University of Florida.

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Lecture 04-2007 - Random Variables and Probability...

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