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...

Let X,. X,... be a sequence of independent identically distributed<br>RVs having common MGF M(0), and let N be an RV taking non-negative integer<br>values with PGF $(s); assume that N is independent of the sequence (X,). Show that<br>Z = X, + X; +...+ X, has MGF 4(M(0)).<br>The sizes of claims made against an insurance company form an independent identically<br>distributed sequence having common PDF f(x)=e-'.x>0. The number of claims during<br>a given year had the Poisson distribution with parameter A. Show that the MGF of the<br>total amount T of claims during the year is<br>(0) = exp{A0/(1 – 0)} for 0 < 1.<br>Deduce that T has mean and variance 2A.<br>

Extracted 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="">

Jun 11, 2022

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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...

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