 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2010/m/m1/b/i/solution

Take

Y = a X_{1} + b

. Directly,

E (e^{t Y}) = e^{b t} E (e^{a t X_{1}}) = e^{b t} M_{1} (a t) .

The formula holds wherever the moment-generating function on the right is finite.

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