Consider (n) independent random variables (U_{1}, U_{2}, ldots, U_{n}), each of which obeys a Cauchy density function, [ p_{U}(u)=frac{1}{pi betaleft[1+left(frac{u}{beta} ight)^{2} ight]} ] where (beta)

Consider \(n\) independent random variables \(U_{1}, U_{2}, \ldots, U_{n}\), each of which obeys a Cauchy density function,

\[ p_{U}(u)=\frac{1}{\pi \beta\left[1+\left(\frac{u}{\beta}\right)^{2}\right]} \]

where \(\beta\) is real and positive.

(a) Show that this density function does not have a finite second moment.

(b) Show that the random variable

\[ Y=\frac{1}{n} \sum_{i=1}^{n} U_{i} \]

obeys a Cauchy distribution for all \(n\) and, therefore, does not obey the central limit theorem.