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For instance, the geometric and exponential distributions have extremely convenient properties of being memoryless. Furthermore, the exponential distribution is perfect for modeling the time between events in a Poisson process. The Poisson distribution itself is wonderful for calculating the number of events in a given time period in a stationary process.

My question is this, are these properties just conveniences or were these distribution sought after originally to model such processes? How did they arise historically so I can better understanding the grounding for them in present text rather than just falling from the sky with convenient properties? This is a strange question but would settle my 'not getting it' feeling for probability. It seems that in all the other branches of mathematics there is more context for how the collective mind proceeded from step 1 to step 10 whereas in statistics my feeling of uncertainty arises since it feels as though everything just fell from the sky.