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

\[f(x)= \begin{cases}1-n^{-1} & x=0 \\ n^{-1} & x=n^{\alpha} \\ 0 & \text { elsewhere }\end{cases}\]

where \(\alpha \in \mathbb{R}\).

a. For what values of \(\alpha\) does \(X_{n} \xrightarrow{p} 0\) as \(n ightarrow \infty\) ?

b. For what values of \(\alpha\) does \(X_{n} \xrightarrow{\text { a.c. }} 0\) as \(n ightarrow \infty\) ?

c. For what values of \(\alpha\) does \(X_{n} \xrightarrow{c} 0\) as \(n ightarrow \infty\) ?