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a) In a sequence of consecutive years 1, 2, ...,T an annual number of bankruptcies are recorded by the Central Bank. The random counts N;, i = 1, 2, ...,T of bankruptcies in a given year are modeled using a Poisson(A) distribution and can be assumed independent from year to year. The Central Bank has collected data for the past eight years and recorded the following counts: 3, 2, 2, 1, 1, 0, 2, 3. The prior on A is a Gamma(3, ). Determine the posterior distribution h(A|N). Note: You may use that for a known a > 0 and 3 > 0, the Gamma(a, 3) density is given by: epa-1 f(x; a, B) x > 0, r(a)Ba where r(a) = e *xª-ldx a-1 is the gamma function. Also, if X is distributed Gamma(a, B) then E(X)= aß and Var(X) = a32.

a) In a sequence of consecutive years 1, 2, ...,T an annual number of bankruptcies are recorded by the Central Bank. The random counts N;, i = 1, 2, ...,T of bankruptcies in a given year are modeled using a Poisson(A) distribution and can be assumed independent from year to year. The Central Bank has collected data for the past eight years and recorded the following counts: 3, 2, 2, 1, 1, 0, 2, 3. The prior on A is a Gamma(3, ). Determine the posterior distribution h(A|N). Note: You may use that for a known a > 0 and 3 > 0, the Gamma(a, 3) density is given by: epa-1 f(x; a, B) x > 0, r(a)Ba where r(a) = e *xª-ldx a-1 is the gamma function. Also, if X is distributed Gamma(a, B) then E(X)= aß and Var(X) = a32.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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a) In a sequence of consecutive years 1, 2, ...,T an annual number of bankruptcies are recorded by the Central Bank. The random counts N;, i = 1, 2, ...,T of bankruptcies in a given year are modeled using a Poisson(A) distribution and can be assumed independent from year to year. The Central Bank has collected data for the past eight years and recorded the following counts: 3, 2, 2, 1, 1, 0, 2, 3. The prior on A is a Gamma(3, ). Determine the posterior distribution h(A|N). Note: You may use that for a known a > 0 and 3 > 0, the Gamma(a, 3) density is given by: epa-1 f(x; a, B) x > 0, r(a)Ba where r(a) = e *xª-ldx a-1 is the gamma function. Also, if X is distributed Gamma(a, B) then E(X)= aß and Var(X) = a32.
Transcribed Image Text:a) In a sequence of consecutive years 1, 2, ...,T an annual number of bankruptcies are recorded by the Central Bank. The random counts N;, i = 1, 2, ...,T of bankruptcies in a given year are modeled using a Poisson(A) distribution and can be assumed independent from year to year. The Central Bank has collected data for the past eight years and recorded the following counts: 3, 2, 2, 1, 1, 0, 2, 3. The prior on A is a Gamma(3, ). Determine the posterior distribution h(A|N). Note: You may use that for a known a > 0 and 3 > 0, the Gamma(a, 3) density is given by: epa-1 f(x; a, B) x > 0, r(a)Ba where r(a) = e *xª-ldx a-1 is the gamma function. Also, if X is distributed Gamma(a, B) then E(X)= aß and Var(X) = a32.
b) The central bank (from part (a)) claims that the intensity A is less than 2. The central bank would like to test the claim via Bayesian testing with a zero-one loss. Determine the value of the posterior probability required to perform this hypothesis test. Give your answer to 2 decimal places. You might want to use the following: Г(п) — (п — 1)! and .2 16--10r de e 0.00016
Transcribed Image Text:b) The central bank (from part (a)) claims that the intensity A is less than 2. The central bank would like to test the claim via Bayesian testing with a zero-one loss. Determine the value of the posterior probability required to perform this hypothesis test. Give your answer to 2 decimal places. You might want to use the following: Г(п) — (п — 1)! and .2 16--10r de e 0.00016
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