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(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].

(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].

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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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. n (a) Write down a function f so that the joint distribution of Y, X₁,..., Xn can be described via ·b P(a ≤ Y ≤ b, X₁ = k1, X2 = k2 … . . Xn = kn) = [ f(y, k₁, k2, ... kn)dy a 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: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. n (a) Write down a function f so that the joint distribution of Y, X₁,..., Xn can be described via ·b P(a ≤ Y ≤ b, X₁ = k1, X2 = k2 … . . Xn = kn) = [ f(y, k₁, k2, ... kn)dy a 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].
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