Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 13I / i

Table of contents
- Solution i

Solution

The density is that of a Gamma distribution with shape

ν > 0

and rate

ν / μ_{i}

, requiring

μ_{i} > 0

. Direct integration gives

E (Y_{i}^{r}) = (\frac{μ _{i}}{ν})^{r} \frac{Γ ( ν + r )}{Γ ( ν )}

for

ν + r > 0

. Using

Γ (ν + 1) = ν Γ (ν)

and

Γ (ν + 2) = ν (ν + 1) Γ (ν)

,

E Y_{i} = μ_{i}, var Y_{i} = μ_{i}^{2} / ν .

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