List of top Probability Questions on Expectation and Moments

Let \( X_i, i = 1, 2, \dots, n \), be i.i.d. random variables with the probability density function \[ f_X(x) = \begin{cases} \frac{1}{\sqrt{2 \Gamma\left( \frac{1}{6} \right)}} x^{-\frac{5}{6}} e^{-\frac{x}{8}}, & 0<x<\infty, \\ 0, & \text{elsewhere}, \end{cases} \] where \( \Gamma(\cdot) \) denotes the gamma function. Also, let \( \bar{X}_n = \frac{1}{n} (X_1 + X_2 + \cdots + X_n) \). If \[ \sqrt{n} \left( \bar{X}_n - \mathbb{E}[\bar{X}_n] \right) \xrightarrow{d} N(0, \sigma^2), \] then \( \sigma^2 \) (rounded off to two decimal places) is equal to ________

Let \( X_i, i = 1, 2, \dots, n \), be i.i.d. random variables from a normal distribution with mean 1 and variance 4. Let \( S_n = X_1^2 + X_2^2 + \dots + X_n^2 \). If \( \text{Var}(S_n) \) denotes the variance of \( S_n \), then the value of \[ \lim_{n \to \infty} \left( \frac{\text{Var}(S_n)}{n} - \left( \frac{E(S_n)}{n} \right)^2 \right) \quad \text{(in integer) is equal to} \, ________. \]