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Poisson(A) Xn - Binomial(n, p) Question 4: What is the marginal distribution of X?

Poisson(A) Xn - Binomial(n, p) Question 4: What is the marginal distribution of X?

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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The inhabitants of a city develop skin cancer at an approximate rate X. For those people who have developed skin cancer, some proportion p E (0, 1) will die from the disease. Assume a simple model such that n, the number of people who develop skin cancer, is distributed Poisson(A). Let X denote the people who die from skin cancer in this city. Then, assuming that every cancer patient is independent of the others, and that the proportion p is constant: ~ U Poisson(A) X\n - Binomial(n, p) Question 4: What is the marginal distribution of X? Hint: two primary ways to do this: 1. you can summate out n from the joint distribution directly (more straightforward, but tricky algebra) o Pay attention to the lower limit of n o Remember that i=0 2. use MGFS + iterated expectation: E[etx] = En|Exn[etx n|| and then recognize the distribution corresponding to the MGF (need to understand MGFS and iterated expectation). o Obtain the inner expectation by using the Binomial theorem. O X Binoтial(n, Ap) We do not have enough information to answer this question. Poisson(Ap) X~ Binomial(4, р) f(æ) = { )p° (1 – p)*-* for 0 < æ < n O.w.
Transcribed Image Text:The inhabitants of a city develop skin cancer at an approximate rate X. For those people who have developed skin cancer, some proportion p E (0, 1) will die from the disease. Assume a simple model such that n, the number of people who develop skin cancer, is distributed Poisson(A). Let X denote the people who die from skin cancer in this city. Then, assuming that every cancer patient is independent of the others, and that the proportion p is constant: ~ U Poisson(A) X\n - Binomial(n, p) Question 4: What is the marginal distribution of X? Hint: two primary ways to do this: 1. you can summate out n from the joint distribution directly (more straightforward, but tricky algebra) o Pay attention to the lower limit of n o Remember that i=0 2. use MGFS + iterated expectation: E[etx] = En|Exn[etx n|| and then recognize the distribution corresponding to the MGF (need to understand MGFS and iterated expectation). o Obtain the inner expectation by using the Binomial theorem. O X Binoтial(n, Ap) We do not have enough information to answer this question. Poisson(Ap) X~ Binomial(4, р) f(æ) = { )p° (1 – p)*-* for 0 < æ < n O.w.
The inhabitants of a city develop skin cancer at an approximate rate A. For those people who have developed skin cancer, some proportion p E (0, 1) will die from the disease. Assume a simple model such that n, the number of people who develop skin cancer, is distributed Poisson(A). Let X denote the people who die from skin cancer in this city. Then, assuming that every cancer patient is independent of the others, and that the proportion p is constant: n ~ Poisson(A) X\n ~ Binomial(n, p) Question 5: What is the distribution of nX? X(1-р) п - XX ~ Віnomial р We do not have enough information to answer this question. O n|X ~ Poisson(Xp) n – X|X ~ Poisson(A(1 – p) ) п — 1 for n > x n|X ~ fn|x (n|x) n! 1- Ei-a i! O.w.
Transcribed Image Text:The inhabitants of a city develop skin cancer at an approximate rate A. For those people who have developed skin cancer, some proportion p E (0, 1) will die from the disease. Assume a simple model such that n, the number of people who develop skin cancer, is distributed Poisson(A). Let X denote the people who die from skin cancer in this city. Then, assuming that every cancer patient is independent of the others, and that the proportion p is constant: n ~ Poisson(A) X\n ~ Binomial(n, p) Question 5: What is the distribution of nX? X(1-р) п - XX ~ Віnomial р We do not have enough information to answer this question. O n|X ~ Poisson(Xp) n – X|X ~ Poisson(A(1 – p) ) п — 1 for n > x n|X ~ fn|x (n|x) n! 1- Ei-a i! O.w.
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