cauchy distribution

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If your eye isn’t quite trained, you might have looked at the above picture and thought, “Oh, a family of normal / Gaussian distributions!” And… you would be wrong. The distributions above are various forms of the Cauchy distribution which has a wider peak and fatter tails than the normal distribution.

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The interesting about the Cauchy distribution though is that although it is a bell-shaped curve its mean and variance are undefined. Due to the equation through which the probability distribution function is defined, it is not possible to calculate the moments needed to get the mean or variance, nor is it possible to calculate any other finite moments. The Cauchy distribution does have many uses (the one I’m most familiar with is it being part of the solution to Laplace’s Equation on the upper half plane which is fascinating in itself), though it stands as a classic pathological example in probability and statistics.

±∞

The Cauchy distribution (?dcauchy in R) nails a flashlight over the number line

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and swings it at a constant speed from 9 o'clock down to 6 o'clock over to 3 o'clock. (Or the other direction, from 3→6→9.) Then counts how much light shone on each number.

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In other words we want to map evenly from the circle (minus the top point) onto the line. Two of the most basic, yet topologically distinct shapes related together.

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You’ve probably heard of a mapping that does something close enough to this: it’s called tan.

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Since tan is so familiar it’s implemented in Excel, which means you can simulate draws from a Cauchy distribution in a spreadsheet. Make a column of =RAND()’s (say column A) and then pipe them through tan. For example B1=TAN(A1). You could even do =TAN(RAND()) as your only column. That’s not quite it; you need to stretch and shift the [0,1] domain of =RAND() so it matches [−π,+π] like the circle. So really the long formula (if you didn’t break it into separate columns) would be =TAN( PI() * (RAND()−.5) ). A stretch and a shift and you’ve matched the domains up. There’s your Cauchy draw.

In R one could draw three Cauchy’s with rcauchy(3) or with tan(2*(runif(3).5)).

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What’s happening at tan(−3π/2) and tan(π/2)? The tan function is putting out to ±∞.

I saw this in school and didn’t know what to make of it–I don’t think I had any further interest than finishing my problem set.

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I saw as well the ±∞ in the output of flip[x]= 1/x.

  • 1/−.0000...001 → −∞ whereas 1/.0000...0001 → +∞.

It’s not immediately clear in the flip[x] example but in tan[x/2] what’s definitely going on is that the angle is circling around the top of the circle (the hole in the top) and the flashlight of the Cauchy distribution could be pointing to the right or to the left at a parallel above the line.

Why not just call this ±∞ the same thing? “Projective infinity”, or, the hole in the top of the circle.

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