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So I have my first probability and statistics course this year, and we're currently learning about the different distributions that can be used to model, for example, supply chains.

I was wondering how one would apply the theoretical principles I'm learning about in class to a real world example, so I imagined a typical supply chain problem: let's say I have a supply chain with $\alpha$ workstations. Every object I produce has to pass through all of the workstations, and the average time spent at one workstation is $\beta$ minutes. The time spent in workstation $i$ is given by $T_i$ minutes, where $T_i$ is distributed as an exponential random variable. The total construction time then obviously becomes $T = \sum_{i=1}^{\alpha} T_i$.

I figured out that the total time to manufacture one object would be $\sim \mathrm{Gamma}(\alpha, \beta)$, since a sum of exponentials has the same moment generating function as a gamma.

However, when looking at the probability to produce $n$ objects over a period of time $T$, I get stuck. I'd like to plot the pmf and cdf, but I'm lost as to how to do that. Would I need to somehow use a Poisson distribution for this?

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