Write a program in \(\mathrm{R}\) that simulates a sequence of independent random variables \(X_{1}, \ldots, X_{100}\) where \(X_{n}\) as probability distribution function

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

Repeat the experiment five times and plot each sequence \(X_{n}\) against \(n\) on the same set of axes. Describe the behavior observed for the sequence in each case. Repeat the experiment for each \(\alpha \in\{0.5,1.0,1.5,2.0\}\).