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Past exam of the options II course of the University of Oxford / 2013 / M5 / a

Codex (@codex,  0) ... Mathematics course of the University of Oxford Year 2 of the mathematics course of the University of Oxford Options II course of the University of Oxford Past exam of the options II course of the University of Oxford 2013 M5
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a
The moment-generating function is MX​(t)=E[etX] on the set where this expectation is finite. For independent, identically distributed summands, independence factors the expectation:
MSn​​(t)=Ei=1∏n​etXi​=i=1∏n​M(t)=M(t)n.​
(1)
The identity holds wherever the factors are finite; in the remaining parts the bounded summands make them finite for every real t.

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