Q 4.4. An individual picked at random from a population has a propensity to have accidents that is modelled by a random variable Y having the gamma distribution with shape parameter a and rate parameter 3. Given Y = y, the number of accidents that the individual suffers in years 1, 2,..., n are independent random variables X₁, X2,... Xn each having the Poisson distribution with parameter y. (a) Write down a function f so that the joint distribution of Y, X₁,..., Xn can be described via P(a ≤ Y ≤ b, X₁ = k1, X₂ = k2₂. . . Xxn = kn) = f f (y, k₁, k2, ... kn) dy and derive from this expression that, for your choice of f, Y has the Gamma distribution, and that conditionally on Y = y, X₁, X2,... Xn are independent, each having the Poisson distribution with parameter y. (b) Find the conditional distribution of Y given that X₁ = k1₁, X2 = k2, ..., kn.
Q 4.4. An individual picked at random from a population has a propensity to have accidents that is modelled by a random variable Y having the gamma distribution with shape parameter a and rate parameter 3. Given Y = y, the number of accidents that the individual suffers in years 1, 2,..., n are independent random variables X₁, X2,... Xn each having the Poisson distribution with parameter y. (a) Write down a function f so that the joint distribution of Y, X₁,..., Xn can be described via P(a ≤ Y ≤ b, X₁ = k1, X₂ = k2₂. . . Xxn = kn) = f f (y, k₁, k2, ... kn) dy and derive from this expression that, for your choice of f, Y has the Gamma distribution, and that conditionally on Y = y, X₁, X2,... Xn are independent, each having the Poisson distribution with parameter y. (b) Find the conditional distribution of Y given that X₁ = k1₁, X2 = k2, ..., kn.
Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter11: Data Analysis And Probability
Section: Chapter Questions
Problem 8CR
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![Q 4.4. An individual picked at random from a population has a propensity to have accidents
that is modelled by a random variable Y having the gamma distribution with shape parameter a
and rate parameter 3. Given Y = y, the number of accidents that the individual suffers in years
1, 2, ..., n are independent random variables X₁, X2,... Xn each having the Poisson distribution
with parameter y.
(a) Write down a function f so that the joint distribution of Y, X₁,..., Xn can be described
via
=ff(y, k₁, k₂..
f(y, k₁, k2, ... kn)dy
P(a ≤ Y ≤ b, X₁ = k₁, X2 = k2 … . . Xn = kn) = [.
..
and derive from this expression that, for your choice of f, Y has the Gamma distribution,
and that conditionally on Y = y, X₁, X2,... Xn are independent, each having the Poisson
distribution with parameter y.
(b) Find the conditional distribution of Y given that X₁ = k₁, X2 = k2, . . . , kn.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb41de797-8c36-43f3-a49e-0d77bbbd163e%2Fb54d9cc7-a444-424f-9ec7-ccaf5ea4575c%2F5xtphbj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Q 4.4. An individual picked at random from a population has a propensity to have accidents
that is modelled by a random variable Y having the gamma distribution with shape parameter a
and rate parameter 3. Given Y = y, the number of accidents that the individual suffers in years
1, 2, ..., n are independent random variables X₁, X2,... Xn each having the Poisson distribution
with parameter y.
(a) Write down a function f so that the joint distribution of Y, X₁,..., Xn can be described
via
=ff(y, k₁, k₂..
f(y, k₁, k2, ... kn)dy
P(a ≤ Y ≤ b, X₁ = k₁, X2 = k2 … . . Xn = kn) = [.
..
and derive from this expression that, for your choice of f, Y has the Gamma distribution,
and that conditionally on Y = y, X₁, X2,... Xn are independent, each having the Poisson
distribution with parameter y.
(b) Find the conditional distribution of Y given that X₁ = k₁, X2 = k2, . . . , kn.
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