(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)
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Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
Section: Chapter Questions
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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].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb41de797-8c36-43f3-a49e-0d77bbbd163e%2Fd0cafa09-d1de-4db8-8d85-987d0c7dc7da%2F5wzpiuc_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.
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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