If someone could please explain how I go about this question as indicated, that would be amazing! Revising for exams and this one has stumped me. From a little bit of research, i think its the Erlang distribution in disguise. Thanks!
2026-03-27 00:56:38.1774572998
Help understanding this question on the Gamma Distribution
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Observe that if
$$X_1\sim Gamma(1;\beta)= exp(\beta)$$
it is easy to prove that $E(X_1)=1/\beta$ thus
$$X\sim Gamma(\alpha;\beta)$$
can always be viewed as a sum of independent exponentials an thus its mean is
$$\frac{\alpha}{\beta}$$
Note that in this parametrization $\beta$ is the rate parameter; if you consider it as the scale parameter the result will be modified in $E(X)=\alpha\cdot\beta$