Question

For $n \in \mathbb{N}$, let $X_1, X_2, \ldots, X_n$ be a random sample from the Cauchy distribution having probability density function
$f(x) = \frac{1}{\pi(1 + x^2)}, \quad -\infty < x < \infty.$
Let $g: \mathbb{R} \to \mathbb{R}$ be defined by
$g(x) = \begin{cases} x, & \text{if } -1000 \leq x \leq 1000 \\\0, & \text{otherwise}\end{cases}.$
Let
$\alpha = \lim_{n \to \infty} P\left( \frac{1}{n^{3/4}} \sum_{i=1}^n g(X_i) > \frac{1}{2} \right).$
Then $100\alpha$ is equal to __________ (answer in integer).

Answer 1

The correct Answer is : 0 -0(Approx.)