Cauchy Distribution

The Cauchy distribution is one of the continuous probability distributions. This distribution has many peculiar properties. This distribution plays an important role in illustrating various theoretical aspects of probability theory and statistics rather and particular applications.

Definition

A random variable (rv) X has a general Cauchy distribution with parametersμ andλ if its probability density function (pdf) is given by

$$f(x) \ = \ \frac{1}{π} \ \frac{λ}{λ^2 + (x-μ)^2} \ \ \ ,\ - \infty \ < \ x \ < \ \infty$$

$$f(x) \ = \ \frac{1}{πλ \left [1 + \left ( \frac{x-μ}{λ} \right )^2 \right ]} \,\ - \infty \ < \ x \ < \ \infty$$

where,μ > 0, andλ > 0.

If X follows a Cauchy bdistribution with parametersμ andλ, we write X ~ C (μ,λ).

The function f(x) is non negative and is a probability density function since

$$\int_{-\infty}^\infty f(x) d x \ = \ \frac{1}{\pi} \\int_{-\infty}^\infty \ \frac{λ}{λ^2 + ( x-μ)^2} dx$$

$$= \ \frac{1}{\pi} \ \int_{-\infty}^\infty \ \frac{dy}{1+y^2} \ \ where \ y \ = \frac{x-\mu}{\lambda} \ \Rightarrow \ dy \ = \ \frac{dx}{\lambda}$$

$$= \ \frac{1}{\pi} \ [{tan^-}^1 y]_{-\infty}^{\infty}$$

$$= \ \frac{1}{\pi} \ \left [ \ \frac{\pi}{2} - \left ( \ - \ \frac{-\pi}{2} \ \right ) \ \right ]$$

$$= \ 1$$

The graph of general Cauchy Distribution is shown in the figure given below. The Cauchy curve is symmetrical about the line X =μ.

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The distribution function of cauchy distribution is given by

f(x) = P (X≤ x)

$$= \ \int_{- \infty}^x f(u) du$$

$$= \ \frac{1}{\pi} \ \int_{-\infty}^x \ \frac{\lambda}{\lambda^2 \ + \ (u - \mu)^2} \ du$$

$$= \ \frac{1}{\pi} \ \int_{-\infty}^{x} \ \frac{dy}{1 + y^2} \ \ \ \ \ \ where \ y \ = \ \frac{u - \mu}{\lambda} \ \Rightarrow \ du \ = \lambda \ dy$$

$$= \ \frac{1}{\pi} \ [{tan^-}^1 y]_{-\infty}^{x}$$

$$= \ frac{1}{\pi} \ \left [ {tan^-}^{1} \ \left ( \frac{x - \mu}{\lambda} \right ) \ + \ \frac{\pi}{2} \ \right ]$$

$$ = \ \frac{1}{2} \ + \frac{1}{\pi} \ {tan^-}^{1} \ \left ( \ \frac{x- \mu}{\lambda} \right )$$

Standard Cauchy Distribution

If X follows a Cauchy distribution with parameters μ and λ then a standardb cauchy variable is defined as follows by

$$Z \ = \ \frac{x - \mu}{\lambda}$$

and its distribution is known as stsndsrd cauchy distribution.

Definition

A random variable Z is said to have a standard cauchy distribution if its probability density function (pdf) is given by

$$f(z) \ = \ \frac{1}{\pi (1 + z^2)}$$

If a random variable X follows a standard Cauchy distribution, we write X ~ C (0,1).

THe graph of the standard Cauchy density function f(z) is symmetrical about the Y-axis, fairly flat with meanμ = 0 as shown in the following figure:

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The cumulative distribution function of the standard Cauchy variable X is given by

F(x) = P(X≤ x)

$$= \ \int_{-\infty}^{\infty} \ f(t)dt$$

$$= \ \frac{1}{\pi} \ \int_{-\infty}^{\infty} \ \frac{1}{1 + t^2} \ dt$$

$$= \ \frac{1}{\pi} \ [{tan^-}^1 y]_{-\infty}^{x}$$

$$= \ \frac{1}{\pi} \ [{tan^-}^{1}x \ - \ {tan^-}^{1}(-\infty)]$$

$$= \ \frac{1}{\pi} \ \left [ \ {tan^-}^{1}x \ - \ \left ( \ - \ \frac{\pi}{2} \right ) \ \right ]$$

$$= \ \frac{1}{2} \ \frac{1}{\pi} \ {tan^-}^{1} x \ \ \ \ \ ; - \infty \ < \ x \ < \ \infty$$

As the standard Cauchy curve y = f(x) is symmetrical about meanμ = 0, the

mean = median = mode = 0.

The variance of the standard Cauchy distribution is infinite. To prove it, let X ~ C (0,1). Then

E (x) = μ = 0.

$$E(x^2) = \ \int_{-\infty}^{\infty} \ x^2 \ f(x) \ dx$$

$$= \ \int_{-\infty}^{\infty} \ \frac{x^2}{\pi (1+x^2)} \ dx$$

$$= \ \frac{2}{\pi} \ \int_{0}^{\infty} \ \frac{x^2}{1+x^2} \ dx$$

$$= \ \frac{2}{\pi} \ \int_{0}^{\infty} \ \frac{1 + x^2 - 1}{1+x^2} \ dx$$

$$= \ \frac{2}{\pi} \ \int_{0}^{\infty} \ \frac{1 + x^2}{1+x^2} \ dx \ -\frac{2}{\pi} \ \int_{0}^{\infty} \ \frac{1}{1+x^2} \ dx$$

$$= \ \frac{2}{\pi} \ \int_{0}^{\infty} \ dx \-\frac{2}{\pi} \ \int_{0}^{\infty} \ \frac{1}{1+x^2} \ dx$$

$$= \ [ \ x \ ]_{0}^{\infty} \ - \ \frac{2}{\pi} \ [{tan^-}^{1} x]_{0}^{\infty}$$

$$ = \ \infty \ - \ \frac{2}{\pi} \ {tan^-}^{1} \infty$$

$$= \ \infty \ - \ \frac{2}{\pi} \ \frac{\pi}{2}$$

$$ = \ \infty \ - \ 1$$

$$= \ \infty$$

since the first integral diverges to infinity.

$$\therefore \ V(X) \ = \ E(X^2) \ - \ [E(X)]^2 \ = \ \infty$$

Moments Of Cauchy Distribution

Let X ~ C (μ,λ ). Then the r^th moments about origin of X is given by

μ_r' = E(X^r)

$$= \ \int_{-\infty}^{\infty} \ x^r \ f(x) \ dx$$

$$= \ \frac{1}{\pi} \ \int_{-\infty}^{\infty} \ x^r \ \frac{\lambda}{\lambda^2 \ + \ ( x - \mu)^2 \ dx$$

$$= \ \frac{\lambda}{\pi} \ \int_{-\infty}^{\infty} \ \frac{x^r}{\lambda^2 + (x-\mu)^2}} \ dx$$

which does not exist because the integral is not convergent. Therefore, for the Cauchy distribution, the moments do not exist. In particualr, the mean and variance of the Cauchy distribution do nop\t exist. So we do not obtain the moment generating function of this distribution.

Bibliography

Sukubhattu N.P. (2013). Probability & Inference - II. Asmita Books Publishers & Distributors (P) Ltd., Kathmandu.

Larson H.J. Introduction to Probability Theory and Statistical Inference. WileyInternational, New York.