Let \( \{X_n\}_{n \geq 1} \) be a sequence of i.i.d. random variables with the common probability density function
\[
f(x) = \frac{1}{\pi(1 + x^2)}, \quad -\infty<x<\infty.
\]
Define
\[
Y_n = \frac{1}{2} + \frac{1}{\pi} \tan^{-1}(X_n) { for } n = 1, 2, \dots.
\]
Then which one of the following options is correct?