One distribution of a random variable is important not for its applications, but for what it tells us about our definitions. The Cauchy distribution is one such example, sometimes referred to as a pathological example. The reason for this is that although this distribution is well defined and has a connection to a physical phenomenon, the distribution does not have a mean or a variance. Indeed, this random variable does not possess a moment generating function.
Definition of the Cauchy Distribution
We define the Cauchy distribution by considering a spinner, such as the type in a board game. The center of this spinner will be anchored on the y axis at the point (0, 1). After spinning the spinner, we will extend the line segment of the spinner until it crosses the x axis. This will be defined as our random variable X.
We let w denote the smaller of the two angles that the spinner makes with the y axis. We assume that this spinner is equally likely to form any angle as another, and so W has a uniform distribution that ranges from -π/2 to π/2.
Basic trigonometry provides us with a connection between our two random variables:
X = tanW.
The cumulative distribution function of Xis derived as follows:
H(x) = P(X < x) = P(tan W < x) = P(W < arctanX)
We then use the fact that W is uniform, and this gives us:
H(x) = 0.5 + (arctan x)/π
To obtain the probability density function we differentiate the cumulative density function. The result is h(x) = 1/π (1 + x2)
Features of the Cauchy Distribution
What makes the Cauchy distribution interesting is that although we have defined it using the physical system of a random spinner, a random variable with a Cauchy distribution does not have a mean, variance or moment generating function. All of the moments about the origin that are used to define these parameters do not exist.
We begin by considering the mean. The mean is defined as the expected value of our random variable and so EX = ∫-∞∞x /π (1 + x2) dx.
We integrate by using substitution. If we set u = 1 +x2 then we see that du = 2x dx. After making the substitution, the resulting improper integral does not converge. This means that the expected value does not exist, and that the mean is undefined.
Similarly the variance and moment generating function are undefined.
Naming of the Cauchy Distribution
The Cauchy distribution is named for the French mathematician Augustin-Louis Cauchy (1789 - 1857). Despite this distribution being named for Cauchy, information regarding the distribution was first published by Poisson.