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The variable of interest may not have a Gaussian distribution, but its expected value and variance generally are. Sure, there's some pathological cases, but th
by xapata 6y ago
The variable of interest may not have a Gaussian distribution, but its expected value and variance generally are. Sure, there's some pathological cases, but the Cauchy distribution doesn't show up that often.
- solveit 6y agoDoes the Cauchy distribution ever actually show up?
- ooobit2 6y agoLess often than meetings start and end on time.
- leephillips 6y agoYes. Check the example section of: http://www.math.nagoya-u.ac.jp/~richard/teaching/s2019/Cauchy_Distribution.pdf http://www.math.nagoya-u.ac.jp/~richard/teaching/s2019/Cauch...
- jointpdf 6y agoIf X, Y ~ N(0,1), then X/Y ~ Cauchy(0,1) (Read: If X and Y are normally distributed random variables with mean 0 and standard deviation 1, then X/Y is equivalent to a Cauchy(0,1) distributed RV. This is a useful equivalence to know if working with standard normal RVs, which is often. E[X/Y] does not exist!) If X ~ Cauchy(0,1), then X ~ normalized student-t distribution. (student-t puts the “t” in t-SNE, where it acts as a weighting function on the Euclidean distance between points. UMAP uses a parameterized version of this weight function that is essentially a generalization of the Cauchy distribution: https://jlmelville.github.io/smallvis/umap.html https://jlmelville.github.io/smallvis/umap.html)