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Let X,. X,... be a sequence of independent identically distributed RVs having common MGF M(0), and let N be an RV taking non-negative integer values with PGF 6(s); assume that N is independent of the sequence (X,). Show that Z= X, + X, +...+ X, has MGF (M(0)). The sizes of claims made against an insurance company form an independent identically distributed sequence having common PDF f(x)=e-'. x20. The number of claims during a given year had the Poisson distribution with parameter A. Show that the MGF of the total amount T of claims during the year is (0) = exp{A0/(1 – 0)} for 0 < 1. Deduce that T has mean A and variance 2A.

Let X,. X,... be a sequence of independent identically distributed RVs having common MGF M(0), and let N be an RV taking non-negative integer values with PGF 6(s); assume that N is independent of the sequence (X,). Show that Z= X, + X, +...+ X, has MGF (M(0)). The sizes of claims made against an insurance company form an independent identically distributed sequence having common PDF f(x)=e-'. x20. The number of claims during a given year had the Poisson distribution with parameter A. Show that the MGF of the total amount T of claims during the year is (0) = exp{A0/(1 – 0)} for 0 < 1. Deduce that T has mean A and variance 2A.

MATLAB: An Introduction with Applications
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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Let X. X,.... be a sequence of independent identically distributed RVs having common MGF M(0), and let N be an RV taking non-negative integer values with PGF $(s); assume that N is independent of the sequence (X,). Show that Z= X, + X2 + ...+ X, has MGF (M(0)). The sizes of claims made against an insurance company form an independent identically distributed sequence having common PDF f(x)=e-', x>0. The number of claims during a given year had the Poisson distribution with parameter A. Show that the MGF of the total amount T of claims during the year is (0) = exp{A0/(1 – 0)} for 0 < 1. Deduce that T has mean A and variance 2A.
Transcribed Image Text:Let X. X,.... be a sequence of independent identically distributed RVs having common MGF M(0), and let N be an RV taking non-negative integer values with PGF $(s); assume that N is independent of the sequence (X,). Show that Z= X, + X2 + ...+ X, has MGF (M(0)). The sizes of claims made against an insurance company form an independent identically distributed sequence having common PDF f(x)=e-', x>0. The number of claims during a given year had the Poisson distribution with parameter A. Show that the MGF of the total amount T of claims during the year is (0) = exp{A0/(1 – 0)} for 0 < 1. Deduce that T has mean A and variance 2A.
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