Papa Charlie Jackson and Blind Blake Part 1 (Blind Blake, October 1929) [Remastered] – YouTube

Papa Charlie Jackson and Blind Blake Part 1 (Blind Blake, October 1929) [Remastered] – YouTube.

Papa Charlie Jackson – The Judge Cliff Davis Blues (1926) Blues Banjo – YouTube

Papa Charlie Jackson – The Judge Cliff Davis Blues (1926) Blues Banjo – YouTube.

Musings on Cauchy Distribution

Cauchy distributions have an “infinte mean” and an “infinite variance”.

Frequentist Perpective:
We think of the sample mean as, if it exists, the value of the observed quantities divided by the number of quantities observed. However, if you are all over the place (due to the fat tails), then no extra information is gained and you do not zero in on the mean. In fact, you are no better off repeating the experiment millions of times than if you did it just once.
This is due to the fact that there is an infinite variance. An infinite variance conjures thoughts of a truly random process, does it not? Something that has an equal probability of being any number on (-inf, inf) (much like a uniform(-inf, inf) has an infinite variance. And sure enough, if you look at the back of the book, the varX of a uniform(a,b) is (b-a)^2/12. so that is (2*inf)^2/12=inf.

Infinite variance actually means something i.e. you are just as likely to get any number. This leads to the idea that there is no mean, we say the mean is infinite, but really it just doesn’t make sense to talk about a mean. It would be more intuitive to say the mean is undefined, since an “infinite mean” suggests you expect your system to return a huge huge number, which is not the case ( you can get pi, or -10^100^100^100, or 500 or whatever)

We can still calculate the median thanks to symmetry.

Physical Analogy:
We usually think of the mean as at the expected value (normal case) or as an average value. If the distribution were made out of wood, and you were to hold it and let it dangle two different ways and drew a line straight down. the intersection of the lines would give you the centroid with location (x,y) where x is the mean of X and y is the mean of Y. Well, if the tails of the distribution as so fat that as you go off many many standard deviations away there is a still a high probability of x, then holding the piece of wood at different locations around the median will make no difference and the wood will stay level. THink in terms of scale: if you balance a pen on your finger (do it right now), it will stay level for infinitely uncountable locations within a region around the median.

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