Past exam of the probability, statistics and options I course of the University of Oxford / 2007 / M / 3

Solution

M_{X} (t) = E e^{tX}

on the real values where this expectation is finite, normally including a neighborhood of zero. Direct substitution gives

M_{a X + b} (t) = e^{b t} M_{X} (a t)

whenever the right side is finite.

For the given Gamma distribution and

t < λ

M_{X} (t) = \frac{λ ^{m}}{( m - 1 )!} \int_{0}^{\infty} x^{m - 1} e^{- (λ - t) x} d x = (\frac{λ}{λ - t})^{m},

using the Gamma function after the scaling

u = (λ - t) x

. At

t \geq λ

the positive integral diverges.

M_{\sum_{i} Y_{i}} (t) = (\frac{λ}{λ - t})^{2 n} .

Uniqueness of a moment-generating function on a neighborhood of zero identifies

i = 1 \sum n Y_{i} \sim Gamma (2 n, λ),

where

λ

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