🚀 go-pugleaf

The 'quiz of the day' got answered on the same and the following days. Many
thanks to those of you who responded (so quickly). As for those who asked for
a summary (and beyond), here it is:

(The quiz, rephrased)

Let P and Q are independent variates of Normal distribution of mean=0 and
standard deviation= Sp and Sq, respectively. Let R=P/Q.
What is the distribution of R?

(The answer)

R will follow Cauchy distribution. The normalized density function is

f(R) = (Sr/_pi_)/(R^2+Sr^2),

where Sr=Sp/Sq and _pi_= 3.1415926535897932..., and

INTEGRATE(from a to b)[f(R) dR] = (1/_pi_)*arctan(R/Sr)|(from a to b).

(The note)

1. Cauchy distribution is the same as Lorentzian distribution. Or may be not,
   but I have done some NMR (sorry, it's Nuclear Magnetic Resonance) spectra
   analysis, by using Lorentzian band to decompose NMR spectra. Both
   distribution have the same density function.
2. The parameter Sr in f(R) is analog to standard deviation in Normal
   distribution. Sr is the half width at half height of the peak.
3. Let P'=P/Sp, Q'=Q/Sq (which imply Sp'=Sq'=1) and R'=P'/Q', then Sr'=1.

(The most-cited reference)

Advanced Theory of Statistics, originally by Sir Maurice Kendall
5th ed., Vol. I, 'Distribution Theory', by Alan Stuart and J. Keith Ord
(Vol. II also contains many useful stuffs)

(The beyond)

The following table contains accumulated probability of R in 8 intervals,
(-inf,-10), (-10,-1), (-1,-0.1), (-0.1,0), (0,0.1), (0.1,1), (1,10), (10,+inf)
from three simulations. Data in each column of 'SIMULATE' are obtained from
10^7 R's, and those of 'INTEGRATE' are calculated according to the integral
expression above.

Accumulated probability of Cauchy distribution of R=P/Q, Sr=Sp/Sq
==============================================================================
         Sp=1, Sq=2, Sr=0.5      Sp=1, Sq=1, Sr=1        Sp=2, Sq=1, Sr=2

         SIMULATE  INTEGRATE     SIMULATE  INTEGRATE     SIMULATE  INTEGRATE
==============================================================================
-inf
         0.016107  0.015902      0.031650  0.031726      0.062803  0.062833
-10.0
         0.131533  0.131681      0.218344  0.218274      0.289752  0.289583
 -1.0
         0.289553  0.289583      0.218205  0.218274      0.131613  0.131681
 -0.1
         0.062802  0.062833      0.031769  0.031726      0.015813  0.015902
  0
         0.062824  0.062833      0.031660  0.031726      0.016073  0.015902
 +0.1
         0.289866  0.289583      0.218351  0.218274      0.131529  0.131681
 +1.0
         0.131609  0.131681      0.218253  0.218274      0.289616  0.289583
+10.0
         0.015706  0.015902      0.031768  0.031726      0.062801  0.062833
+inf
==============================================================================

Ping

PS. By the time I'm ready to post this, 9 more responses come in with more
    suggestions, hints, references, answers and requests. You boys and girls
    are really beautiful. Thanks again.