Discussion
If a random variable X has Cauchy distribution with location \(\mu\) and scale \(\sigma\), then the distribution of \(Y = \dfrac{1}{X}\) is :
Answer: B
Y is Cauchy with location \(\mu/(\mu^{2} + \sigma^{2})\) and scale \(\sigma/(\mu^{2} + \sigma^{2})\).
Change variables with \(x = 1/y\), \(|dx/dy| = 1/y^{2}\):
\[ \begin{aligned} f_{Y}(y) &= \frac{\sigma/\pi}{y^{2}\left(\sigma^{2} + (1/y - \mu)^{2}\right)} \\ &= \frac{\sigma/\pi}{(\mu^{2} + \sigma^{2})y^{2} - 2\mu y + 1} \end{aligned} \]
Completing the square with \(m = \mu/(\mu^{2} + \sigma^{2})\) and \(s = \sigma/(\mu^{2} + \sigma^{2})\), the denominator is \((\mu^{2} + \sigma^{2})[(y - m)^{2} + s^{2}]\), so \(f_{Y}(y) = \dfrac{s/\pi}{(y - m)^{2} + s^{2}}\), the Cauchy density with location \(m\) and scale \(s\).
With \(\mu = 0, \sigma = 1\) this is the familiar fact that the reciprocal of a standard Cauchy variable is standard Cauchy. The other options keep \(\mu\) or \(\sigma\) unchanged, which fails as soon as \(\mu \ne 0\), or deny that Y is Cauchy.
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