mont4e_sm_ch07_supplemental

01 then f x1 x 2 2 e x1 x2 001e 001

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Unformatted text preview: close to the MLE of the mean. f ( x | λ ) = λe− λx , x ≥ 0 and f (λ ) = 0.01e −0.01λ . Then, f ( x1 , x 2 , λ ) = λ2 e − λ ( x1 + x2 ) 0.01e −0.01λ = 0.01λ 2 e − λ ( x1 + x2 + 0.01) . As a function of λ, this is recognized as a gamma density with parameters 3 and x1 + x 2 + 0.01 . Therefore, the posterior mean for λ is ~ λ= 3 3 = = 0.00133 . x1 + x2 + 0.01 2 x + 0.01 1000 b) Using the Bayes estimate for λ, P(X<1000)= ∫ 0.00133e −.00133 x dx =0.736. 0 Supplemental Exercises n 7-45 f ( x1 , x 2 ,..., x n ) = ∏ λe −λxi for x1 > 0, x 2 > 0,..., x n > 0 i =1 7-15 7-46 ⎛1 f ( x1 , x 2 , x3 , x 4 , x5 ) = ⎜ ⎜ ⎝ 2π σ 7-47 f ( x1 , x 2 , x 3 , x 4 ) = 1 7-48 X 1 − X 2 ~ N (100 − 105, 7-49 5 5 2 ⎞ i ⎟ exp⎛ − ∑ ( x2− μ2 ) ⎞ ⎜ ⎟ ⎟ ⎝ i =1 σ ⎠ ⎠ X ~ N (50,144 ) for 0 ≤ x1 ≤ 1,0 ≤ x 2 ≤ 1,0 ≤ x 3 ≤ 1,0 ≤ x 4 ≤ 1 1.52 2 2 +) 25 30 ~ N (−5,0.2233) P (47 ≤ X ≤ 53) = P⎛ 4...
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