Lecture04-2010 - Conditional Probability and Distribution...

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Conditional Probability and Distribution Functions: Lecture IV Charles B. Moss August 27, 2010 I. Conditional Probability and Independence A. In order to defne the concept oF a conditional probability it is necessary to defne 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 di±erent rolls oF the die where the value oF the frst 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 =4is 1
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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 ±rst 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 forgoing example, the case of the fair die, this value is 1/6.
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Lecture04-2010 - Conditional Probability and Distribution...

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