For example, the standard deviation of a random variable that follows a Cauchy distribution is undefined because its expected value " ? " is undefined.
42.
However, due to the fat tails of the Cauchy distribution, the efficiency of the estimator decreases as more of the sample gets used in the estimate.
43.
Stable distributions have 0 < ? d " 2, with the upper bound corresponding to the normal distribution, and ? = 1 to the Cauchy distribution.
44.
Where \ gamma is the scale factor and \ mu is the peak position of the " unwrapped " distribution . characteristic function of the Cauchy distribution yields:
45.
An infinitesimal formula for an infinitely tall, unit impulse delta function ( infinitesimal version of Cauchy distribution ) explicitly appears in an 1827 text of Augustin Louis Cauchy.
46.
Since the normal distribution, the Cauchy distribution, and the L�vy distribution all have the above property, it follows that they are special cases of stable distributions.
47.
The Cauchy distribution is often used in statistics as the canonical example of a " moments of order greater than or equal to one; only fractional absolute moments exist.
48.
Note also that this implies that the Pearson type VII distribution subsumes the standard Student's " t "-distribution and also the standard Cauchy distribution.
49.
Clearly, the normal distribution is stable, but there are also other stable distributions, such as the Cauchy distribution, for which the mean or variance are not defined.
50.
McCullagh also wrote, " The distribution of the first exit point from the upper half-plane of a circular Cauchy distribution-- > " circular Cauchy distribution ".
