MATH 464 HOMEWORK 9 SPRING 2016 The following assignment is to be turned in on Thursday, April 28, 2016. 1. Let X be a normal random variable with parameters R and > 0. Find the cdf and pdf for Z the standardization of X. 2. Let X1 , X2 , , Xn be independent, identically distributed (i.i.d) random variables. Let and 2 be the common mean and variance respectively. Define n n 1X 1 X 2 (Xj X n ) where X n = Xj Y = n1 n j=1 j=1 Find the mean of Y in terms of n, , and 2. 3. Flip a fair coin until you get 100 heads. Use the central limit theorem to find (approximately) the probabilities it takes at least 200, 250, and 300 flips. Hint: Let X be the number of flips to get 100 heads. Write X as the sum of 100 i.i.d. random variables. 4. Let X1 , X2 , , Xn be independent random variables each having the standard normal distribution. a) Find (approximately) P 80 100 X Xj2 120 j=1 b) Find c so that 100 X P Xj2 100 < c = 0.95 . j=1 1