This note introduces four analytical probability distributions and the underlying uncertain processes from which they can be derived. It demonstrates that, when we can build a reasonable model of an uncertain process, we can use the model to derive the probability distribution from fundamental principles and forgo the chore of estimating the distribution subjectively by assessing fractiles. The probability distributions discussed are the binomial, normal, Poisson, and exponential. The associated processes discussed include counting, accumulation, Poisson, and memoryless. The note emphasizes that, although these probability distributions have many legitimate applications, they are applicable only when the specific assumptions about the underlying processes are satisfied--that is, when the uncertain quantity is obtained from a process similar to the one used to drive the probability distribution in the first place.
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