Approximating the Binomial via the CLT] Suppose that X Binomial(n, p). Then, X is approximately Poisson for large n and small p. Since the binomial is a sum of Bernoulli, the central limit theorem (CLT) suggests that the binomial is also approximately normal for large n, provided p is sufficiently large. In this problem, you will study when to adopt one approximation over the other. (a) Apply the CLT to develop a normal approximation to the binomial X Y where Y N (a, b) for some a and b. Check that your answer is consistent with the explicit analysis provided in the CLT lecture. Your approximation should approximate the distribution of X directly, not its standardization. (b) A normal distribution is supported on all real numbers, while a Binomial is only sup- ported on nonnegative integers between 0 and n. It is reasonable to use a normal approximation to the binomial if: (a) n is large enough so that the skew in the binomial is small and the sum involved in computing the binomial CDF can be approximated with an integral, and (b) the normal approximation does not assign significant probability to the events Y < 0 or Y > n. You worked on (a) in problem 2 of HW 5. For this HW, find a condition on n and p that, if satisfied, guarantees that Pr(Y < 0) < 0.01 and Pr(Y > n) < 0.01. You may use facts about the standard normal CDF or the Chernoff bound to introduce your constraints