1. Let S = 1X; be the loss, where N is the frequency distribution and X is the severity distribution. The mgf of S is M¸(t) = M¸(In Mx (t)) where MN and Mx are the mgfs of N and X respectively. a. Prove, by differentiating Ms using the chain rule and known properties of mgfs, that E(S) = E(N) E(X) and var (S) = E(N)var(X) + var (N)E(X)². Hint: for the second derivative (fgh)'=f'gh+fg'h+fgh' b. Now let X be Exponential with mean ẞ and let N be Poisson with rate parameter λ. Give the form of Ms as a function of t (i.e. substitute for the two mgfs), and show that Ms(t) = λẞMs(t)(1 − ẞt) -². Hence or otherwise, derive the mean and variance of S using this mgf. c. Again let X be Exponential with mean ẞ and let N be Poisson with rate parameter λ. Evaluate the mean and variance of S using the formulae given in (a), and compare to those found in (b).
1. Let S = 1X; be the loss, where N is the frequency distribution and X is the severity distribution. The mgf of S is M¸(t) = M¸(In Mx (t)) where MN and Mx are the mgfs of N and X respectively. a. Prove, by differentiating Ms using the chain rule and known properties of mgfs, that E(S) = E(N) E(X) and var (S) = E(N)var(X) + var (N)E(X)². Hint: for the second derivative (fgh)'=f'gh+fg'h+fgh' b. Now let X be Exponential with mean ẞ and let N be Poisson with rate parameter λ. Give the form of Ms as a function of t (i.e. substitute for the two mgfs), and show that Ms(t) = λẞMs(t)(1 − ẞt) -². Hence or otherwise, derive the mean and variance of S using this mgf. c. Again let X be Exponential with mean ẞ and let N be Poisson with rate parameter λ. Evaluate the mean and variance of S using the formulae given in (a), and compare to those found in (b).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:1. Let S = 1X; be the loss, where N is the frequency distribution and X is the
severity distribution. The mgf of S is M¸(t) = M¸(In Mx (t)) where MN and
Mx are the mgfs of N and X respectively.
a. Prove, by differentiating Ms using the chain rule and known
properties of mgfs, that E(S) = E(N) E(X) and var (S) = E(N)var(X) +
var (N)E(X)².
Hint: for the second derivative (fgh)'=f'gh+fg'h+fgh'
b. Now let X be Exponential with mean ẞ and let N be Poisson with rate
parameter λ. Give the form of Ms as a function of t (i.e. substitute for
the two mgfs), and show that Ms(t) = λẞMs(t)(1 − ẞt) -². Hence or
otherwise, derive the mean and variance of S using this mgf.
c. Again let X be Exponential with mean ẞ and let N be Poisson with rate
parameter λ. Evaluate the mean and variance of S using the formulae
given in (a), and compare to those found in (b).
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