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 Incoming links: Gamma density

Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 7 / b / i / Solution 2026-10-25

Use the gamma density and the normalization integral for shape

n - 1

:

E [Y^{- 1}] = \frac{θ ^{n}}{( n - 1 )!} \int_{0}^{\infty} y^{n - 2} e^{- θ y} d y = \frac{θ ^{n}}{( n - 1 )!} \frac{( n - 2 )!}{θ ^{n - 1}} = \frac{θ}{n - 1} .

The same argument, valid for

n \geq 3

, gives

E [Y^{- 2}] = θ^{2} / [(n - 1) (n - 2)]

.

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