 Solution

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

Take

Y = U X_{1}

, where

U

is uniform on

(0, 1)

and independent of

X_{1}

. Conditional expectation gives

E (e^{t U X_{1}}) = \int_{0}^{1} E (e^{u t X_{1}}) d u = \int_{0}^{1} M_{1} (u t) d u .

For

t

in a sufficiently small neighborhood of zero, finiteness follows from finiteness of the original moment-generating function in such a neighborhood.

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