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note11b - S6880#11b Sampling from Marginal Densities An...

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S6880 #11b Sampling from Marginal Densities, An Example
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An Example Outline 1 An Example An Example—Binomial-Beta-Poisson Model Hatching Insect Eggs Example in Full Form (WMU) S6880 #11b S6880, Class Notes #11b 2 / 7
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An Example Binomial-Beta-Poisson Model Hatching Insect Eggs The joint distribution of X , Y , and N considered by Casella and George, 1992: for given a > 0, b > 0 and λ > 0, f ( x , y , m ) ( m x ) y x + a - 1 ( 1 - y ) m - x + b - 1 e - λ λ m m ! , x = 0 , 1 , · · · , m , 0 y 1 , m = 1 , 2 , · · · has practical applications. Conditioning on m and y , x is the number of successful hatchings from m insect eggs, where each egg has success probability y . The values of m and y vary across insects which are modeled in their respective distributions. The marginal density of X is a typical number of successful hatchings among all insects. Suppose we are interested in some characteristics of the marginal distributions, say f X ( x ) and f Y ( y ) and we need to generate samples from the marginal p.d.f. The exact form of the former is nontrivial.
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