Lecture#09-10.pdf - Contents 21 Marginal Distributions 2...

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Contents 21 Marginal Distributions 2 21.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 21.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 22 Conditional Distributions 4 22.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 22.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 23 Independence 7 23.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 23.2 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 23.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Dirichlet Distribution There is a generalization of the Beta distribution to more than one random variable. Suppose ( Y 1 , Y 2 , Y 3 ) represent proportions or probabilities such that Y 1 + Y 2 + Y 3 = 1 . Examples include Y i is the proportion of ingredient i in a mixture of three ingredients. Y i is the proportion of a sample that fall in category i , when there are a total of 3 categories. For example, the number of people in a sample who (1) don’t exercise, (2) exercise up to twice a week, and (3) exercise more than twice a week. Notice, that because of the constraint Y 3 = 1 - Y 1 - Y 2 is no longer random once the bivariate random variable ( Y 1 , Y 2 ) has been defined. Thus, there are really only two random variables in the collection ( Y 1 , Y 2 , Y 3 ) . A joint continuous pdf for ( Y 1 , Y 2 ) is given by f ( y 1 , y 2 ) = 360 y 2 1 y 2 (1 - y 1 - y 2 ) 0 y 1 , y 2 1 , y 1 + y 2 1 0 otherwise ( Note. The conditions that appear on the right-hand-side are now very important as they indicate all con- straints on the random variable. If you ignore them, you set your limits of integration incorrectly, and get incorrect answers.) The above pdf is a specific example of the generalization of the Beta ( α, β ) distribution to two dimensions. The generalized Beta distribution is called the Dirichlet distribution (if you want to look it up, or remember it later). The random variable ( Y 1 , Y 2 ) lives on the triangle, with vertices { (0 , 0) , (1 , 0) , (0 , 1) } . There are a number of questions I could ask you about this distribution (listed below). If you have questions about any of these, be sure to ask. Verify that f ( y 1 , y 2 ) defines a pdf. In other words, show Z 1 0 Z 1 - y 1 0 360 y 2 1 y 2 (1 - y 1 - y 2 ) dy 2 dy 1 = 1 Find the constant (in this case given already as 360 ) such that f ( y 1 , y 2 ) is a pdf. What is the probability that Y 1 0 . 5 and Y 2 0 . 21 ? What is the probability that Y 1 + Y 2 0 . 5 ?
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21 Marginal Distributions 21.1 Definitions Definition : marginal pmf/pdf If Y 1 , Y 2 are discrete random variables with joint pmf p ( y 1 , y 2 ) , then the marginal pmf’s are p Y 1 ( y 1 ) = X y 2 p ( y 1 , y 2 ) p Y 2 ( y 2 ) = X y 1 p ( y 1 , y 2 ) If Y 1 , Y 2 are continuous random variables with joint pdf f ( y 1 , y 2 ) , then the marginal pdf’s are f Y 1 ( y 1 ) = Z -∞ f ( y 1 , y 2 ) dy 2 f Y 2 ( y 2 ) = Z -∞ f ( y 1 , y 2 ) dy 1 21.2 Examples Discrete Example I A simplified version of an old example is 300 red beads, 200 blue beads mixed in a box. You reach in and randomly select with replacement 3 beads. For each red bead, you earn $1, and for each blue bead, you win $2. Let R be the number of red beads you select, and W be the winnings you earn. In this case, W is a deterministic function of R , namely W = R + 2(3 - R ) = 6 - R but generally, there will be additional randomness in W . The sample space for the experiment is S = { ( r, r, r ) , ( r, r, b ) , ( r, b, r ) , ( b, r, r ) , ( r, b, b ) , ( b, r, b ) , ( b, b, r ) , ( b, b, b ) } The probability of each outcome can be determined by recognizing the independence of each draw: P ((
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