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Suppose we observe iid data y₁,..., yn from Poisson distribution with parameter A. Let A have the Gamma(a, 3) distribution, the conjugate prior distribution for the Poisson likelihood, where a and 3 are known prior parameters. (a) Find the posterior distribution for A. Now, an ecologist counts the numbers of centipedes in each of n = 20 twenty one- metre-square quadrats. The numbers y₁,..., y20 are in the second column labelled as y in the dataset. (b) Let the last three digits of your ID number be ABC. Suppose we want the prior mean for A to be 5+ A and the prior standard deviation to be 5+ B. Find the prior distribution parameters that satisfy this. (c) Using the prior distribution from (b), find the posterior distribution for A. (d) Calculate the posterior median and a 95% credible interval for X. (e) Calculate the posterior median and a 95% credible interval for 0, where 0 = 1 - exp(-X). X 2.256 1.83 2.061 1.108 1.287 1.696 1.518 0.984 0.868 1.387 Y 14 13 7 10 15 15 2 13 13 11 10 13 5 13 9 12 9 12 8 7

Suppose we observe iid data y₁,..., yn from Poisson distribution with parameter A. Let A have the Gamma(a, 3) distribution, the conjugate prior distribution for the Poisson likelihood, where a and 3 are known prior parameters. (a) Find the posterior distribution for A. Now, an ecologist counts the numbers of centipedes in each of n = 20 twenty one- metre-square quadrats. The numbers y₁,..., y20 are in the second column labelled as y in the dataset. (b) Let the last three digits of your ID number be ABC. Suppose we want the prior mean for A to be 5+ A and the prior standard deviation to be 5+ B. Find the prior distribution parameters that satisfy this. (c) Using the prior distribution from (b), find the posterior distribution for A. (d) Calculate the posterior median and a 95% credible interval for X. (e) Calculate the posterior median and a 95% credible interval for 0, where 0 = 1 - exp(-X). X 2.256 1.83 2.061 1.108 1.287 1.696 1.518 0.984 0.868 1.387 Y 14 13 7 10 15 15 2 13 13 11 10 13 5 13 9 12 9 12 8 7

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
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for part b please show the method using example values for A,B,C. the dataset is shown on the side

 

Suppose we observe iid data y₁,..., yn from Poisson distribution with parameter A. Let A have the Gamma(a, 3) distribution, the conjugate prior distribution for the Poisson likelihood, where a and 3 are known prior parameters. (a) Find the posterior distribution for X. Now, an ecologist counts the numbers of centipedes in each of n = 20 twenty one- metre-square quadrats. The numbers y₁, ..., y20 are in the second column labelled as y in the dataset. (b) Let the last three digits of your ID number be ABC. Suppose we want the prior mean for A to be 5 + A and the prior standard deviation to be 5+ B. Find the prior distribution parameters that satisfy this. (c) Using the prior distribution from (b), find the posterior distribution for A. (d) Calculate the posterior median and a 95% credible interval for X. (e) Calculate the posterior median and a 95% credible interval for 0, where 0 = 1 - exp(-X). X 42 у 2.256 14 1.83 13 7 2.061 1.108 10 1.287 1.696 15 1.518 2 13 0.984 0.868 13 1.387 11 43 050 50 === ܘ ܗ ܗ 15 LL115 10 13 WOH 13 9 9 HOHOHOOT 12 12 8 7
Transcribed Image Text:Suppose we observe iid data y₁,..., yn from Poisson distribution with parameter A. Let A have the Gamma(a, 3) distribution, the conjugate prior distribution for the Poisson likelihood, where a and 3 are known prior parameters. (a) Find the posterior distribution for X. Now, an ecologist counts the numbers of centipedes in each of n = 20 twenty one- metre-square quadrats. The numbers y₁, ..., y20 are in the second column labelled as y in the dataset. (b) Let the last three digits of your ID number be ABC. Suppose we want the prior mean for A to be 5 + A and the prior standard deviation to be 5+ B. Find the prior distribution parameters that satisfy this. (c) Using the prior distribution from (b), find the posterior distribution for A. (d) Calculate the posterior median and a 95% credible interval for X. (e) Calculate the posterior median and a 95% credible interval for 0, where 0 = 1 - exp(-X). X 42 у 2.256 14 1.83 13 7 2.061 1.108 10 1.287 1.696 15 1.518 2 13 0.984 0.868 13 1.387 11 43 050 50 === ܘ ܗ ܗ 15 LL115 10 13 WOH 13 9 9 HOHOHOOT 12 12 8 7
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