Write a program in \(\mathrm{R}\) that generates a sample of size \(b\) from a specified distribution \(F_{n}\) (specified below) that weakly converges to a distribution \(F\) as \(n ightarrow \infty\). Compute the Kolmogorov distance between \(F\) and the empirical distribution from the sample. Repeat this for \(n=5,10,25,50,100,500\), and 1000 with \(b=10,000\) for each of the cases given below. Discuss the behavior of the Kolmogorov distance and \(n\) becomes large in terms of the result of Theorem 4.7 (Pólya). Be sure to notice the specific assumptions that are a part of Theorem 4.7.

a. \(F_{n}\) corresponds to a \(\operatorname{Normal}\left(n^{-1}, 1+n^{-1}ight)\) distribution and \(F\) corresponds to a \(\operatorname{Normal}(0,1)\) distribution.

b. \(F_{n}\) corresponds to a \(\operatorname{Gamma}\left(1+n^{-1}, 1+n^{-1}ight)\) distribution and \(F\) corresponds to a \(\operatorname{Gamma}(1,1)\) distribution.

c. \(F_{n}\) corresponds to a \(\operatorname{Binomial}\left(n, n^{-1}ight)\) distribution and \(F\) corresponds to a POISson(1) distribution.

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Theorem 4.7 (Plya). Suppose that {F} is a sequence of distribution functions such that FF as no where F is a continuous distribution function. Then lim sup F(t) F(t)| = 0. n00 TER