Sums of Random Variables, Random Walks and the Central Limit Theorem
Why do we care about sums of random variables? The answer is that everywhere around us the processes that we see often depend on the accumulation of many contributions or are the result of many effects. The pressure in a room, measured on a surface, is the sum of the order of 1023 momentum exchanges between the air molecules and the surface. The large time tectonic deformation is the (tensorial) sum of the deformation associated with the myriad of earthquakes. Errors and/or uncertainties in measurements are often the aggregation of many sources and are in many cases distributed according to a Gaussian law (see below). In fact, it is hard to find an observation that is not controlled by many variables. Studying the sum of random variables allows us to grasp the fundamental notion of collective behavior without the need for further complications.
KeywordsFractal Dimension Random Walk Central Limit Theorem Step Length Continuous Limit
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