NOTES ON STATISTICS, PROBABILITY and MATHEMATICS

Cauchy distribution:

The function \(f(x) = \frac{1}{1+x^2}\) is the fundamental building block of the Cauchy distribution.In probability theory, the Cauchy distribution is a continuous probability distribution that is often used as a “pathological” example because it lacks a defined mean or variance.

The function \(f(x) = \frac{1}{1+x^2}\) is proportional to the density of the standard Cauchy distribution. However, for a function to be a valid probability density function, the total area under the curve must equal \(1.\) If we integrate your function over all real numbers:

\[\int_{-\infty}^{\infty} \frac{1}{1+x^2} \, dx = [\arctan(x)]_{-\infty}^{\infty} = \frac{\pi}{2} - (-\frac{\pi}{2}) = \pi\]

Because the integral is \(\pi\), we must multiply by a normalization constant of \(\frac{1}{\pi}\) to get the Standard Cauchy Distribution:

\[f(x; 0, 1) = \frac{1}{\pi(1+x^2)}\]

Key Relationships and Properties:

  • The “Witch of Agnesi”: The curve \(y = \frac{a^3}{x^2+a^2}\) is a famous geometric curve known as the Witch of Agnesi. The Cauchy PDF is essentially a scaled version of this curve.

  • Student’s t-distribution: The standard Cauchy distribution is a special case of the Student’s t-distribution with exactly one degree of freedom (\(\nu = 1\)).

  • Heavy Tails: Unlike the normal (Gaussian) distribution, which decays exponentially (\(e^{-x^2}\)), the Cauchy distribution decays polynomially (\(\frac{1}{x^2}\)). This gives it “heavy tails,” meaning extreme outliers are much more likely to occur.

  • The Mean Paradox: If you try to calculate the expected value (mean) of a Cauchy random variable:

\[\int_{-\infty}^{\infty} \frac{x}{\pi(1+x^2)} \, dx\]

the integral is undefined because it results in \(\infty - \infty\). Consequently, the Cauchy distribution has no mean, variance, or higher-order moments.

A common way to generate a Cauchy distribution is to imagine a person standing at the point \((0, 1)\) on a Cartesian plane, spinning around, and firing a laser beam at a random angle \(\theta\) (uniformly distributed between \(-90^\circ\) and \(90^\circ\)). The position \(x\) where the laser hits the x-axis (\(y=0\)) follows a Cauchy distribution. This is because \(x = \tan(\theta)\), and the derivative of the inverse tangent leads directly back to the function \(f(x) = \frac{1}{1+x^2}\):

\[\frac d{dx}\arctan(x)=\frac1{1+x^2}\] It is the derivative precisely because a Probability Density Function (PDF) represents the rate of change of probability over an interval.

The Cauchy distribution \(f(x;x_{0},\gamma )\) is the distribution of the \(x\)-intercept of a ray issuing from \((x_{0},\gamma )\) with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero.

The relationship with the Witch of Agnesi:

From Wikipedia:

To construct this curve, start with any two points O and M, and draw a circle with OM as diameter. For any other point A on the circle, let N be the point of intersection of the secant line OA and the tangent line at M. Let P be the point of intersection of a line perpendicular to OM through A, and a line parallel to OM through N. Then P lies on the witch of Agnesi. The witch consists of all the points P that can be constructed in this way from the same choice of O and M. It includes, as a limiting case, the point M itself.

If \(M=(0,1)\) the curve has obeys the equation

\[f(x)=\frac1{1+x^2}\]

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NOTE: These are tentative notes on different topics for personal use - expect mistakes and misunderstandings.