R help - Aug 2007 - OT: distribution of a pathological random variate

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

On Wed, Aug 29, 2007 at 10:39:17AM -0700, 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.
I think this is the standard Cauchy distribution:

http://en.wikipedia.org/wiki/Cauchy_distribution


-- 
Daniel Lakeland
dlakelan at street-artists.org
http://www.street-artists.org/~dlakelan

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.

See also ?dt

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.

--------------------------------------------------------------------
E-Mail: (Ted Harding) <Ted.Harding at manchester.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 29-Aug-07                                       Time: 19:02:32
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On Wed, 29 Aug 2007, 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)
>

Instead of asking this off topic question here try googling

 	'gaussian ratio'
> 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
>
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>
Charles C. Berry                            (858) 534-2098
                                             Dept of Family/Preventive Medicine
E mailto:cberry at tajo.ucsd.edu	            UC San Diego
http://famprevmed.ucsd.edu/faculty/cberry/  La Jolla, San Diego 92093-0901