Advanced calculus is necessary to prove rigorously the main properties of the Cauchy distribution. It is well known that the Cauchy distribution can be generated by a tangent transformation of the uniform distribution. By interpreting this transformation on a circle, it is possible to present elementary and intuitive proofs of some important and useful properties of the distributio

Cuadras, C. M.

Cuadras Avellana, Carlos María

Secretaría de Estado de Cultura

Geometrical understanding of the Cauchy distribution

Revistes Catalanes amb Accés Obert

QU¨ESTIIO´, vol. 26, 1-2, p. 283-287, 2002
GEOMETRICAL UNDERSTANDING OF THE
CAUCHY DISTRIBUTION
C. M. CUADRAS
Universitat de Barcelona
Advanced calculus is necessary to prove rigorously the main properties
of the Cauchy distribution. It is well known that the Cauchy distribution
can be generated by a tangent transformation of the uniform distribution.
By interpreting this transformation on a circle, it is possible to present
elementary and intuitive proofs of some important and useful properties of
the distribution.
Keywords: Cauchy distribution, distribution on a circle, central limit theo-
rem
AMS Classification (MSC 2000): 60-01; 60E05
– Received December 2001.
– Accepted April 2002.
283
The Cauchy distribution is a good example of a continuous stable distribution for which
mean, variance and higher order moments do not exist. Despite the opinion that this dis-
tribution is a source of counterexamples, having little connection with statistical practi-
ce, it provides a useful illustration of a distribution for which the law of large numbers
and the central limit theorem do not hold. Jolliffe (1995) illustrated this theorem with
Poisson and binomial distributions, and he pointed out the difficulties in handling the
Cauchy (see also Lienhard, 1996). In fact, one would need the help of the characteristic
function or to compute suitable double integrals. It is shown in this article that these
properties can be proved in an informal and intuitive way, which may be useful in an
intermediate course.
If U is a uniform random variable on the interval I    π 2π 2, it is well known that
Y   tanU follows the standard Cauchy distribution, i.e., with probability density
f y   1
π
1
1 y2
 ∞  y  ∞
Also Z   tannU, where n is any positive integer, has this distribution.
The tangent transformation can be described using the geometric analogy of a rota-
ting diameter of a circle (Figure 1). Suppose, using usual rectangular co-ordinates, that
the circle has center O and radius 1, and suppose a diameter POP   of the circle is a
needle which rotates uniformly round the circle. Suppose P is the endpoint with po-
sitive value for x, and let OP make angle U with the x-axis. Then U lies in the inter-
val I    π 2π 2. Let Y   tanU; Y has the Cauchy distribution. Note also that
Y   tanU π, so we only need to look at the half-right part of the final position of the
needle. In the following, we will take all angles in the interval I modulo π and use the
property that if V is any arbitrary random value, then U V (mod π) is also uniform on
the interval I and is independent of U .
U
B
P
P 
A
Q
R
1
1
0
Y
X
X   cotU 
Y   tanU 
x   1
y   1
Figure 1.
284
In Figure 1, Y is the vertical coordinate of the point Q, where Q is the intersection of the
line through the half-needle OP and the line x   1. An interesting physical illustration
is obtained by taking the origin as a radioactive source of α-particles impacting on a
fixed line (Rao, 1973, p. 169).
The use of the circle and the rotating needle can give intuitive demonstrations of several
features of the Cauchy distribution. Some examples are:
1) Y has heavy tails, i.e., Y takes extreme values with high probability.
This feature empirically distinguishes Y from the standard normal and many other
distributions. It is easily seen by observing that an angle U close to π 2 (or  π 2),
which will give a large tangent value Y , has the same probability density   1 π as
an angle close to 0.
2) X   Y1 is also distributed as a standard Cauchy distribution.
By symmetry, using the rotating needle analogy, the Cauchy variable could as well
be generated by using the angle between the needle and the y-axis and projecting
on the line y   1 (giving the value BR in Figure 1). But this is the same as taking
X   cotU   1 Y .
3) The mean does not exist.
Actually, it is easy to give an analytic proof of this result, as taking the expectation
of Y gives an indeterminate integral. Using the analogy of the circle, we should
consider the position of the needle after several successive rotations. It is clear that
the mean position might be anywhere around the circle, and a formal «mean» is not
clearly defined.
4) The distribution of the mean of any number of independent observations has the
same distribution as Cauchy. Consequently, the law of the large numbers describing
the convergent behavior of the mean does not hold for this distribution.
The analogy in 3) can be used again. Rotating the needle several times, we obtain
axes uniformly distributed around the circle, and the intuitive average axis is also
uniformly distributed. Taking U as its angle with the x-axis and constructing Y  
tanU gives the standard Cauchy distribution. A rigorous definition of the mean
direction and a proof of its uniform distribution needs, of course, more advanced
arguments.
The above analogy is clearly not a proof of 4), because the tangent of the angle of
the mean axis is not the mean of the tangents. Instead, the Cauchy distribution for
the mean of the Cauchy sample can be illustrated using the following simulation.
Choose an integer m, generate independent uniform angles u 1    um and compute
u¯   atan
 
1
m
m
∑
i 1
tanui

285
Repeat this operation n times and plot a histogram of the obtained sample u¯ 1     u¯n.
The uniform distribution of u¯ will be quite apparent (Figure 2), showing that tanu¯ is
Cauchy. Note that to recognize the Cauchy distribution by a histogram of a Cauchy
sample is less evident, due to the distortion produced by the very large values.
This geometrical approach, and the associated trigonometry, can give the proof of
other properties, for example:
5) Z   Y1 Y  2 has the standard Cauchy distribution.
This is a consequence of the trigonometric identity cotu  tanu 2   cot2u.
6) If W is any random variable then T   Y W  1 YW  has the standard Cauchy
distribution.
Using the formula tanuv  tanu tanv 1  tanutanv, write W   tanV ,
where V is any angle in I. Then U V (mod π) is uniformly distributed and T  
tanU V.
7) If U1U2 are independently uniform on I, and if Y1   tanU1, Y2   tanU2, and
Y3   tanU1 U2, then Y1Y2Y3 are pairwise independent with standard Cauchy
distributions but are jointly dependent.
From the trigonometric relation in 6) above, we have Y3    Y1 Y2 1 Y1Y2.
Thus the relation Y1Y2Y3   Y1 Y3 exists and the Y -values are not independent.
Further formulas were proved in a similar way by Jones (1999) for the normal and
Cauchy distribution. See also Cuadras (2000). For example:
8) Using that tanU is distributed as tannU for n   234, then if Z is standard
Cauchy so is
2Z 1 Z2 Z 3 Z2 1 3Z2 and 4Z 1 Z2 1 6Z2Z4
9) Using tan2U  c or equivalently combining 2Z 1  Z 2 and Z B 1 BZ,
where B is independent of Z (see above), then
2ZB1 Z2
1 B2 2BZ
is also standard Cauchy.
Finally, while it is relatively easy to prove, using only geometry, that the sum of in-
dependent N(0,1) is also normal (see Mantel, 1972), it is an open question to give a
geometric but conscientious proof, elementary enough for teaching purposes, (i.e., wit-
hout using double integrals or the characteristic function), that the mean of the tangents
of uniform angles is the tangent of an angle also uniformly distributed in I, i.e, following
the Cauchy distribution. See Cohen (2000).
286
000
020
040
060
080
100
120
 2  1 5  1  0 5 0 0 5 1 1 5 2
Figure 2. Histogram of a sample of u¯ revealing a uniform distribution, indicating that tanu¯ has
the standard Cauchy distribution.
APPENDIX
MATLAB code to generate samples giving the histogram of Figure 2.
m   100; n   1000; rand(‘seed’,2002);
for i   1 : n, u   randm1 pi  pi 2;c  tanu;mci   atanmeanc; end
histmc
REFERENCES
Cohen, M. P. (2000). «Cuadras, C. M. (2000) Letter to the Editor, The American Sta-
tistician, 54, 87», The American Statistician, 54, 326.
Cuadras, C. M. (2000). «Distributional relationships arising from simple trigonometric
formulas» (Revisited), The American Statistician, 54, 87.
Jolliffe, I. T. (1995). «Sample sizes and the central limit theorem: The Poisson distri-
bution as an illustration», The American Statistician, 49, 269.
Jones, M. C. (1999). «Distributional relationships arising from simple trigonometric
formulas», The American Statistician, 53, 99-102.
Lienahrd, C. W. (1996). «Sample sizes and the central limit theorem: The Poisson dis-
tribution as an illustration» (Revisited). The American Statistician, 50, 282.
Mantel, N. (1972). «A property of certain distributions», The American Statistician, 26,
29-30.
Rao, C. R. (1973). Linear Statistical Inference and its Applications, New York: Wiley.
287


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