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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 P(a ≤ Y ≤ b, X₁ = k1, X2 = k2 . . . Xn = 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.

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 = k2 . . . Xn = 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.

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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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 ß. 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 = k2 … . . Xn = kn) = [° ƒ (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.
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 ß. 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 = k2 … . . Xn = kn) = [° ƒ (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 = k₁, X2 = k2, ..., kn. (c) An insurance company has observed the number of accidents that an individual has suffered on each of n years and wishes to predict the number of accidents that individual will experience in the next year. To this end let Xn+1 be a further random variable so that, conditionally on Y y, X1, X2,... Xn, Xn+1 are independent, each having the Poisson distribution with parameter y. Write down the value of the conditional expectation E[Xn+1 X1, X2, ... Xn, Y], - and hence determine E[Xn+1|X1, X2, ... Xn].
Transcribed Image Text:(b) Find the conditional distribution of Y given that X₁ = k₁, X2 = k2, ..., kn. (c) An insurance company has observed the number of accidents that an individual has suffered on each of n years and wishes to predict the number of accidents that individual will experience in the next year. To this end let Xn+1 be a further random variable so that, conditionally on Y y, X1, X2,... Xn, Xn+1 are independent, each having the Poisson distribution with parameter y. Write down the value of the conditional expectation E[Xn+1 X1, X2, ... Xn, Y], - and hence determine E[Xn+1|X1, X2, ... Xn].
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