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Question 2 i=1,...,n. Assume that y Let y; be the number of transactions made by customer i in a year, Poisson (2), 2 > 0 is unknown, i = 1,...,n. ~ (a) What is the likelihood function, p(y | λ), for 2 provided by the data y = (y₁,...,yn)? (b) Define the Jeffreys prior, p,(0), for a scalar parameter 0, and derive its form from the Poisson likelihood for this example. (c) Assuming the Jeffreys prior, p,(e), for 0, what is the posterior distribution for ? Suppose now we use a Gamma distribution, Gamma(a, b), as a prior for 2 where a > 0 and b 0 are known. (d) What is the posterior distribution, p(λ | y), for λ under the Poisson model and the prior Gamma(a, b) distribution. (e) Suppose ab. For an uninformative prior, do we need a large or small value for a? If we choose a small value for a, what is the effect on the posterior distribution for compared to a large a?

Question 2 i=1,...,n. Assume that y Let y; be the number of transactions made by customer i in a year, Poisson (2), 2 > 0 is unknown, i = 1,...,n. ~ (a) What is the likelihood function, p(y | λ), for 2 provided by the data y = (y₁,...,yn)? (b) Define the Jeffreys prior, p,(0), for a scalar parameter 0, and derive its form from the Poisson likelihood for this example. (c) Assuming the Jeffreys prior, p,(e), for 0, what is the posterior distribution for ? Suppose now we use a Gamma distribution, Gamma(a, b), as a prior for 2 where a > 0 and b 0 are known. (d) What is the posterior distribution, p(λ | y), for λ under the Poisson model and the prior Gamma(a, b) distribution. (e) Suppose ab. For an uninformative prior, do we need a large or small value for a? If we choose a small value for a, what is the effect on the posterior distribution for compared to a large a?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 2 i=1,...,n. Assume that y Let y; be the number of transactions made by customer i in a year, Poisson (2), 2 > 0 is unknown, i = 1,...,n. ~ (a) What is the likelihood function, p(y | λ), for 2 provided by the data y = (y₁,...,yn)? (b) Define the Jeffreys prior, p,(0), for a scalar parameter 0, and derive its form from the Poisson likelihood for this example. (c) Assuming the Jeffreys prior, p,(e), for 0, what is the posterior distribution for ? Suppose now we use a Gamma distribution, Gamma(a, b), as a prior for 2 where a > 0 and b 0 are known. (d) What is the posterior distribution, p(λ | y), for λ under the Poisson model and the prior Gamma(a, b) distribution. (e) Suppose ab. For an uninformative prior, do we need a large or small value for a? If we choose a small value for a, what is the effect on the posterior distribution for compared to a large a?
Transcribed Image Text:Question 2 i=1,...,n. Assume that y Let y; be the number of transactions made by customer i in a year, Poisson (2), 2 > 0 is unknown, i = 1,...,n. ~ (a) What is the likelihood function, p(y | λ), for 2 provided by the data y = (y₁,...,yn)? (b) Define the Jeffreys prior, p,(0), for a scalar parameter 0, and derive its form from the Poisson likelihood for this example. (c) Assuming the Jeffreys prior, p,(e), for 0, what is the posterior distribution for ? Suppose now we use a Gamma distribution, Gamma(a, b), as a prior for 2 where a > 0 and b 0 are known. (d) What is the posterior distribution, p(λ | y), for λ under the Poisson model and the prior Gamma(a, b) distribution. (e) Suppose ab. For an uninformative prior, do we need a large or small value for a? If we choose a small value for a, what is the effect on the posterior distribution for compared to a large a?
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