M Inbox bhaney 151gmai x Bb Welcome Fall 2015STAT x httpscom

M Inbox bhaney 151@gmai x Bb Welcome Fall 2015-STAT x https://compass2g.illinois. x CChegg Study l Guided Soli x The Voice Voting NBC. x C https:// 2g.illino edu rid-19818935 1/cou es/stat 400 120158 143450/Assignment%2010.pdf compass Exercises 1. Sums of iid random variables are asymptotically normal Let X. Xn be independent and identically distributed (iid) random variables (that have a finite variance). According to the Central Limit Theorem (CLT), X LYLIXE is approximately normal when n is large, in the sense that the cdf of converges as n oc to D, the cdf of a standard normal distribution, N(0, 1) (a) Show that an equivalent way to express the CLT is to say that StLIXi is ap- proximately normal when n is large, in the sense that the cdf of EX converges to 4 as n oo. (b) Use (a) and Ex. 4 in HW 9 in order to show that the following random variables are approximately normal when n is large: i. Y Binom(n, p), iii. Y n Neg. B p), iv. Y Gamma(n, a (c) Compute approximately the probability that you need to toss a fair coin more than 120 times in order to observe 40 Tails. (d) Exercise 12 from Section 5.6. Remark: Part b(i) shows that the Normal approximation to the Binomial is a special case of the CLT. Search the web and Windows 8:18 PM 11/9/2015

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