Use the results on generating functions and limit theorems which can be found here to answer the following questions.

1. Let \(Y_n\) be uniform on the set  \(\{1,2,3,\cdots,n\}\). Find the moment generating function of \(\frac1n Y_n\) which we will call it \(M_n(t)\).  Then show that as \(n \rightarrow \infty\),
   \[ M_n(t) \rightarrow \frac{e^t -1}{t}\]
   Lastly, identify this limiting moment generating function that of a known random variable. Comment on why this make sense.
2. Let \(X_n\) be distributed as a binomial with parameters \(n\) and \(p_n=\lambda/n\).   By using the probability generating function for \(X_n\), show that \(X_n\) converges to a Poisson random variable with parameter \(\lambda\) as \(n \rightarrow \infty\).

[Adapted from Stirzaker, p 318]