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

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- Solution b

Solution

Independence makes the moment-generating function of the sum the product of the two moment-generating functions. For

t < λ

,

M_{X_{1} + X_{2}} (t) = (\frac{λ}{λ - t})^{r_{1}} (\frac{λ}{λ - t})^{r_{2}} = (\frac{λ}{λ - t})^{r_{1} + r_{2}} .

The expectations are finite on, for example,

(- λ /2, λ /2)

, as is the moment-generating function of the candidate Gamma distribution. The uniqueness theorem for moment-generating functions therefore applies, proving

X_{1} + X_{2} \sim Gamma (r_{1} + r_{2}, λ) .

This is additivity of independent gamma distributions with a common rate; a common rate is essential to this specific formula.

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