Write a program in \(\mathrm{R}\) that simulates a sample of size \(b\) from a Bino\(\operatorname{MIAL}\left(n, n^{-1}ight)\) distribution and a sample of size \(b\) from a POISson(1) distribution. For each sample, compute the relative frequencies associated with each of the values \(\{1, \ldots, n\}\). Notice that it is possible that not all of the sample from the POISSON distribution will be used. Run this simulation for \(n=10,25,50\), and 100 with \(b=10,000\) and discuss how well the relative frequencies from each sample compare in terms of the theoretical result given in Example 4.4.

Example 4.4. This example motivates the POISSON approximation to the BINOMIAL distribution. Consider a sequence of random variables {X}1 where X, has a BINOMIAL (n, n) distribution where A is a positive con- stant. Let x be a positive real value. The distribution function of X, is given by for n.