In the textbook, there's a distribution like the following,

$S=\sum_{i=}^{200}X_i\sim Gamma(\alpha = 200, \beta)$

then the textbook define a new function $P$ obtained by diving the $\beta$, so something like:

$W=\frac{S}{\beta}\sim Gamma(\alpha = 200, 1)$

But why?? Are we allow to do this and keep it as a gamma distribution?

Edit: Suppose $P(S< 200\beta)$, will this equal to $P(W < 200\beta )$ or $P(W < 200)$ ??


Yes you can, when Gamma distribution is still in the same form if you (positively) scale it. Intuitively, exponential distribution can be thought of time needed for an event to occur, where Gamma can be thought of the same event occurring $\alpha$ times. The scaling we do here acts like just a unit change, e.g. seconds to hours.

For your second question, you just substitute their definitions: $$P(S<200\beta)=P\left(\frac{S}{\beta}<200\right)=P(W<200)$$

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