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Unformatted text preview: Conditional Probability and Distribution Functions: Lecture IV Charles B. Moss August 27, 2010 I. Conditional Probability and Independence A. In order to define the concept of a conditional probability it is necessary to define joint and marginal probabilities. 1. The joint probability is the probability of a particular combi nation of two or more random variables. 2. Taking the role of two die as an example, the probability of rolling a 4 on one die and a 6 on the other die is 1/36. 3. There are 36 possible outcomes of the two die { 1 , 1 } , { 1 , 2 } , ··· { 2 , 1 } , { 2 , 2 } , ··· { 6 , 6 } . 4. Therefore the probability of a { 4 , 6 } given that the die are fair is 1 / 36. B. The marginal probability is the probability one of the random variables irrespective of the outcome of the other variable. 1. Going back to the die example, there are six different rolls of the die where the value of the first die is 4 { 4 , 1 } , { 4 , 2 } , { 4 , 3 } , { 4 , 4 } , { 4 , 5 } , { 4 , 6 } (1) 2. Hence, again assume that the die are fair the marginal prob ability of x 1 = 4 is 1 AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture IV Fall 2010 P [ x 1 = 4] = P [ { 4 , 1 } ] + P [ { 4 , 2 } ] + P [ { 4 , 3 } ]+ P [ { 4 , 4 } ] + P [ { 4 , 5 } ] + P [ { 4 , 6 } ] = 1 36 + 1 36 + 1 36 + 1 36 + 1 36 + 1 36 = 6 36 = 1 6 (2) C. The conditional probability is then the probability of one event, such as the probability that the first die is a 4, given that the value of another random variable is known, such as the fact that the value of the second die roll is equal to 6. In the forgoingthe value of the second die roll is equal to 6....
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This note was uploaded on 07/15/2011 for the course AEB 6182 taught by Professor Weldon during the Fall '08 term at University of Florida.
 Fall '08
 Weldon

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