1. Let X, X be exchangeable so that the X, are conditionally independent given a parameter...

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1. Let X₁, X be exchangeable so that the X, are conditionally independent given a parameter 0. Suppose that X. | 0 is geometrically distributed with probability density function ; = 1,2,...... (a) Show that f(: | 0), where a = (₁,...,n), belongs to the 1-parameter exponential family. Hence, or otherwise, find the conjugate prior distribution and corresponding posterior distribution for 0. (b) Show that the posterior mean for can be written as a weighted average of the prior mean of and the maximum likelihood estimate, 2-¹ f (x | 0) (1 - 0)-¹0, = 1. Let X₁, X be exchangeable so that the X, are conditionally independent given a parameter 0. Suppose that X. | 0 is geometrically distributed with probability density function ; = 1,2,...... (a) Show that f(: | 0), where a = (₁,...,n), belongs to the 1-parameter exponential family. Hence, or otherwise, find the conjugate prior distribution and corresponding posterior distribution for 0. (b) Show that the posterior mean for can be written as a weighted average of the prior mean of and the maximum likelihood estimate, 2-¹ f (x | 0) (1 - 0)-¹0, =

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a To show that fx belongs to the 1parameter exponential family we can express the probability densit... View the full answer