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.

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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 <0, Fn(x) = P(X x) = (7) (4)* (1 - 4) for 0xn for > n.