Analytical Probability Distributions

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. Excerpt UVA-QA-0437 ANALYTICAL PROBABILITY DISTRIBUTIONS As a U.S. manufacturer of components for consumer electronic products in this highly competitive era, you are very concerned about product quality. You know that, on average, 5% of all components produced by one particular machine are defective, and you are wondering whether you need to replace the machine. To help answer that question, you want to know the probability of a defect-free production run of 10 components. One way to approach this problem is to use your judgment to make a subjective assessment of the probability distribution of the number of defects in a production run of 10 components. Unless you have a great deal of experience with similarly sized production runs, however, you may not feel particularly comfortable with the results. An alternative approach is to reflect on the underlying process by which the number of defects will be determined. When the underlying process that generates the uncertain quantity is well understood, the probability distribution can often be determined from basic principles. Consider a regular tetrahedron. What are the chances that it will land on any particular face when fairly tossed? Once we determine that a regular tetrahedron has four faces of equal area, most of us will answer 25% without hesitation. Because few of us have much experience tossing tetrahedrons, and none of us, presumably, resorted to the formal probability-assessment techniques that we learned in class, how then did we make this assessment? . . .