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 Incoming links: Moment-generating function of an independent product

Past exam of the options I course of the University of Oxford / 2010 / M / M1 / b / iv / Solution 2026-10-31

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.

This uses the moment-generating function of an independent product.

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