Poisson(A) Xn - Binomial(n, p) Question 4: What is the marginal distribution of X?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F419820c1-e9cd-469e-aa42-67c8a6edd9ca%2Ff914462f-399a-467e-832b-e7ff9e794f5b%2F1gdy1o9_processed.png&w=3840&q=75)
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.

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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