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, + X; +...+ X, has MGF 4(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.

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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, + X; +...+ X, has MGF 4(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 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, + X; +...+ X, has MGF 4(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 and variance 2A.
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