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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

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Why is it so cool to square numbers (in terms of finding the standard deviation)?

I think that is quite obvious. Firstly, the sampling standard deviation measures the dispersion of the data in the samples. You square the difference is to reflect how much the actually data is away …
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