[R] OT: distribution of a pathological random variate

Horace Tso Horace.Tso at pgn.com
Wed Aug 29 21:46:55 CEST 2007

```Thank you Ted, Daniel, and Charles. I thought Cauchy distribution has to do with resonance. Didn't know this nice extension.

H.

>>> Ted Harding <Ted.Harding at manchester.ac.uk> 8/29/2007 11:02 AM >>>
On 29-Aug-07 17:39:17, Horace Tso wrote:
> Folks,
>
> I wonder if anything could be said about the distribution of a random
> variate x, where
>
> x = N(0,1)/N(0,1)
>
> Obviously x is pathological because it could be 0/0. If we exclude this
> point, so the set is {x/(0/0)}, does x have a well defined
> distribution? or does it exist a distribution that approximates x.
>
> (The case could be generalized of course to N(mu1, sigma1)/N(mu2,
> sigma2) and one still couldn't get away from the singularity.)
>
> Any insight or reference to related discussion is appreciated.
>
> Horace Tso

A good question -- but it has a long-established answer. X has the
Cauchy distribution, whose density function is

f(x) = 1/(pi*(1 + x^2))

Have a look at ?dcauchy

It is also the distribution of t with 1 degree of freedom.

You don;t need to exclude the point (0,0) explicitly, since
it has zero probabilityof occurring. But the chance that the
denominator could be small enough to give a very large value
of X is quite perceptible.

Try

X<-rcauchy(1000)
max(X)

and similar. Play around!

Best wishes,
ted.

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Date: 29-Aug-07                                       Time: 19:02:32
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