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.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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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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