I have asked this question yesterday, but I lost my account. Could anybody help me with this problem one more time? I want the answer to the first part of this problem. The Chernoff Inequality is in the second picture. And I also post another question for the second part of this problem. Please find it in my profile. Thank you.

WLLN vs CLT and Large vs Small Deviations A stronger version (called Berry-Esseen Theorem) of CLT states that if X1, X2, ..., Xn are independent and identically distributed random variables with zero mean and unit variance and E[X,|3] = 3, then for all r and n we have [Fr(x) - D(x)| 0.47483 Vn 2, where Fr(.) is the CDF of - _ Xi, and o(.) is the CDF of the zero-mean and unit-variance Gaussian random variable. The said bound is due to Irina Shevtsova. A fair coin is tossed 1000 times. Let Pm denote the probability of observing more than m HEADS. . Find an upper bound Am on Pm based on the Chernoff Inequality (WLLN) developed in HW 6, Prob. 7 for m > 500. . Find an upper bound Bm on Pm in terms of o(.) based on Berry-Esseen Theorem (CLT) stated above for m > 500. . Plot the two bounds as a function of m for in the range from 501 to 600. Which bound is better in what region?Consider Chernoff inequality. (a) Let X1, X2, ..., Xn denote n independent Bernoulli random variables with parameter p. Let Z = _ _, Xi. Use Chernoff inequality and show the following for any 8 > 0: P(Z > p+8)