3.8 The Cauchy Distribution: $\mathrm{𝖢𝖺𝗎𝖼𝗁𝗒}$

1. $f_X(x)=\frac{1}{\pi (1+x^2)}$ for ,

2. $F_X(x)=\frac{1}{\pi }\arctan (x)+\frac{1}{2}$,

3. $𝖤\left[X\right]$ not defined.

We write $X\sim \mathrm{𝖢𝖺𝗎𝖼𝗁𝗒}$.

Unnumbered Figure: Link

Now, for $b>0$,

and similarly, for , ${\int }_a^0\frac{x}{\pi (1+x^2)}𝑑x=-\log (1+a^2)$. Thus

which is not defined since it can take any desired value depending on the relative speeds with which $a\to -\mathrm{\infty }$ and $b\to \mathrm{\infty }$.

Convolution: If $X_1,\mathrm{\dots },X_n$ are independent Cauchy rvs, then $(X_1+\mathrm{\dots }+X_n)/n\sim \mathrm{𝖢𝖺𝗎𝖼𝗁𝗒}$.

Transformations: If $U\sim \mathrm{𝖴𝗇𝗂𝖿}(-\pi /2,\pi /2)$, then $\tan (U)\sim \mathrm{𝖢𝖺𝗎𝖼𝗁𝗒}$. If $X_1\sim 𝖭(0,1)$ and $X_2\sim 𝖭(0,1)$ are independent, then $X_1/X_2\sim \mathrm{𝖢𝖺𝗎𝖼𝗁𝗒}$.

Reciprocal: if $X\sim \mathrm{𝖢𝖺𝗎𝖼𝗁𝗒}$ then $1/X\sim \mathrm{𝖢𝖺𝗎𝖼𝗁𝗒}$. Quiz: How do we know this, given the above? When $X_1$ and $X_2$ are both $𝖭(0,1)$, $X_1/X_2$ and $X_2/X_1$ must have the same distribution, by symmetry.