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According to this post I read

http://www.clayford.net/statistics/deriving-the-gamma-distribution/

the gamma distribution $\text{Gamma}(\alpha,\lambda)$ is the theoretical distribution of wait times until the $\alpha$-th change for a Poisson process, where the average number of events in a given period is $\lambda$.

Therefore, since $\chi^2_k$ is a special case of the gamma distribution, i.e.$$\chi^2_k \sim \text{Gamma}\bigg(\frac{k}{2},\frac{1}{2}\bigg)$$ we can interpret it as the theoretical distribution of wait times until the $\frac{k}{2}$-th change for a Poisson process, where the average number of events in a given period is $\frac{1}{2}$.

However, I am used to defining the chi-squared distribution as $Z_i \sim \mathcal{N}(0,1)$ $$\chi_k^2 \sim \sum_{i=1}^k Z_i ^2$$

How do I interpret the sum of $Z_i$ in terms of wait time? What do the $Z_i$ tell me about how long I will have to wait?

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