(a) Let $X_{1}, \ldots, X_{n}$ be identically distributed random variables with a common probability density function $F$ such that $E\left(x_{1} ight)=\mu$ and $\operatorname[Var}\left(x{1} ight)=\sigma {2} \ino, \infty) $. Evaluate (i) $\left(n^{-1} \sum_{j=1}^{n} X_{j} ight)$ and (ii) $\operatorname[Var}\left(n^{-1} \sum_{j=1}^{n} X_{j} ight)$. (b) If we impose, in addition, that $X$ 's are independent as well, using Chebyshev's inequality, or otherwise, prove the weak law of large numbers: For any slepsilon>$, $$ \lim _{n ightarrow \infty! \operatorname Pr}\left(\left|\bar{X}_{n}- Imu ight|>\epsilon ight)=0, $$ where $\bar{X}_{n} $ denotes the sample mean, i.e. sbartx_{n}