Proving X Follows a Cauchy Distribution if 1/X Follows a Cauchy Distribution

Understanding the relationship and properties of the Cauchy distribution can be crucial in various statistical analyses. This article will demonstrate how to prove that if a random variable Y 1/X follows a Cauchy distribution, then the random variable X also follows a Cauchy distribution.

Properties of the Cauchy Distribution

The Cauchy distribution is characterized by its heavy tails and the lack of a defined mean or variance. A random variable X is said to follow a standard Cauchy distribution if its probability density function (PDF) is given by:

f_X(x) frac{1}{pi(1 x^2)}

Transformation of Random Variables

If Y 1/X has a Cauchy distribution, then we aim to find the distribution of X by examining the relationship between X and Y.

Find the CDF of Y

The cumulative distribution function (CDF) of a Cauchy random variable Y is given by:

F_Y(y) frac{1}{2} frac{1}{pi} tan^{-1}(y)

Transforming to Find the CDF of X

Since Y 1/X, we can find X in terms of Y as:

X 1/Y

To find the CDF of X, we express it in terms of Y:

F_X(x) P(X ≤ x) P(1/Y ≤ x) P(Y ≥ 1/x) 1 - P(Y

Now substituting the CDF of Y:

F_X(x) 1 - left(frac{1}{2} frac{1}{pi} tan^{-1}(1/x)right)

Simplifying this gives:

F_X(x) frac{1}{2} - frac{1}{pi} tan^{-1}(1/x)

Find the PDF of X

To find the PDF of X, we differentiate the CDF:

f_X(x) frac{d}{dx} F_X(x) -frac{1}{pi} frac{d}{dx} tan^{-1}(1/x)

Using the chain rule:

frac{d}{dx} tan^{-1}(1/x) frac{-1/x^2}{1 1/x^2} frac{-1}{x^2 1}

Thus:

f_X(x) -frac{1}{pi}left(-frac{1}{x^2 1}right) frac{1}{pi(1 x^2)}

This is exactly the PDF of a standard Cauchy distribution.

Conclusion

Therefore, if Y 1/X follows a Cauchy distribution, then X must also follow a Cauchy distribution. This can be formally proven by deriving the PDF of X from the CDF of Y, confirming that X retains the properties of the Cauchy distribution.