Consider a sequence of independent random variables \(\left\{X_{n}ight\}_{n=1}^{\infty}\) where \(X_{n}\) has probability distribution function

\[f_{n}(x)= \begin{cases}2^{-(n+1)} & x=-2^{n(1-\varepsilon)}, 2^{n(1-\varepsilon)} \\ 1-2^{-n} & x=0 \\ 0 & \text { elsewhere }\end{cases}\]

where \(\varepsilon>\frac{1}{2}\) (Sen and Singer, 1993).

a. Compute the mean and variance of \(X_{n}\).

b. Let

\[\bar{X}_{n}=n^{-1} \sum_{k=1}^{n} X_{k}\]

for all \(n \in \mathbb{N}\). Compute the mean and variance of \(\bar{X}_{n}\).

c. Prove that \(\bar{X}_{n} \xrightarrow{p} 0\) as \(n ightarrow \infty\).