Past exam of the probability course of the University of Oxford / 2014 / 2 / a

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- Solution a

Solution

Integrating

e^{t x}

against the Gamma density gives the moment-generating function of a gamma distribution:

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

The change of variables

y = (λ - t) x

reduces the integral to

Γ (r) = (r - 1)!

using the gamma function. For

t \geq λ

the defining integral is infinite. In particular the moment-generating function is finite on a neighbourhood of zero.

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