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Limit theorems via generating functions
Use the results on generating functions and limit theorems which can be found here to answer the following questions.
- 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. - 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]
Basic generating functions
In each example below find the probability generating function (p.g.f.) or moment generating function (m.g.f.) of the random variable \(X\): (show your work!) ( When both are asked for, recall the relationship between the m.g.f. and p.g.f. You only need to do the calculation for one to find both.)
- \(X\) is normal mean \(\mu\) and variance \(\sigma^2\). Find m.g.f.
- \(X\) is uniform on \( (0,a)\). Find m.g.f.
- \(X\) is Bernoulli with parameter \(p\). Find p.g.f. and m.g.f.
- \(X\) is exponential with parameter \(\lambda\). Find m.g.f.
- \(X\) is geometric with parameter \(p\). Find p.g.f. and m.g.f.
- \(X=a+bY\) where \(Y\) has probability generating function \(G(s)\). Find m.g.f.
Random Sum of Random Variables
Let \(\{X_r : r=1,2,3,\cdots\}\) be a collection of i.i.d. random variables. Let \(G(s))\) be the generating function of \(X_1\) ( i.e. \(G(s)=\mathbf{E} (s^{X_1})\) ), and hence; each of the \(X_r\)’s. Let \(N\) be an additional random variable taking values in the non-negative integers which is independent of all of the \(X_r\). Let \(H(s)\) be generating function of \(N\).
- Define the random variable \[ T=\sum_{k=1}^N X_k\] where \(T=0\) of \(N=0\). For any fixed \(s>0\), calculate \( \mathbf{E}[ s^T | N]\). Show that the generating function of \(T\) is \(H(G(s)) \).
- Assume that each claim that a given insurance company pays is independent and distributed as an exponential random variable with parameter \(\lambda\). Let the number of claims in a given year be distributed as geometric random variable with parameter \(p\). What is the moment generating function of the total amount of money payed out in a given year ? Use your answer to identify the distribution of the total money payed out in a given year.
- Looking back at the previous part of the question, contrast your answer with the result of adding a non random number of exponential together.
