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hwk5_soln

# hwk5_soln - IEOR E4702 Statistical Inference for Financial...

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IEOR E4702 Statistical Inference for Financial Engineering HWK Solution 5 1. (a) The likelihood for X 1 ,. . . , X n is given by f ( X | λ )= n Y i =1 e λ λ X i X i ! . The prior density is λ α 1 e λ / β Γ ( α ) β α . Therefore, the posterior is given by f ( λ | X ) λ α 1 e λ / β Γ ( α ) β α · n Y i =1 e λ λ X i X i ! λ α 1 e λ / β · n Y i =1 e λ λ X i = λ α 1+ P n i =1 X i exp ½ μ 1 β + n λ ¾ . Thus, the posterior density f ( λ | X ) must be a gamma with parameter ˜ α and ˜ β ,where ˜ α = α + n X i =1 X i , ˜ β = μ 1 β + n 1 . (b) Since the log-likelihood is log f ( X | λ n λ + n X i =1 X i log λ + n X i =1 log( X i !) , we have log f ( X | λ ) ∂λ = n + X λ 2 log f ( X | λ ) 2 = X λ 2 . Therefore, I ( λ E 2 log f ( X | λ ) 2 ¸ = λ λ 2 = 1 λ . 1

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Thus, the Je f erys’ prior is f ( λ ) 1 λ . The posterior density is given by f ( λ | X ) 1 λ · n Y i =1 e λ λ X i X i !
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hwk5_soln - IEOR E4702 Statistical Inference for Financial...

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