# Connecting interpretations of chi-squared distribution as both gamma distribution and normal distribution

According to this post I read

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?