Unformatted text preview: STAT 100A HWVI Due next Wed
Problem 1: Suppose we flip a fair coin n times independently. Let X be the number of heads. Let k = n/2 + z n/2, or z = (k - n/2)/( n/2). Let g(z) = P (X = k). 2 (1) Using the Stirling formula n! 2nnn e-n , show that g(0) 1 n . a b means that 2 a/b 1 as n . 2 (2) Show that g(z)/g(0) e-z /2 as n . (3) For two integers a < b, let a = (a - n/2)/( n/2), and b = (b - n/2)/( n/2). Show that 2 b P (a X b) a f (z)dz, where f (z) = 1 e-z /2 . 2 (4) Let Z = (X - n/2)/( n/2). Show that P (a X b) = P (a Z b ). Argue that in the limit Z N(0, 1). Problem 2: Suppose Z N(0, 1). Calculate E[Z] and Var[Z]. Problem 3: Suppose we flip a fair coin 1000 times independently. Let X be the number of heads. (1) What is the probability that 480 X 520? (2) What is the probability that X > 530? Problem 4: Suppose among the population of voters, 1/3 of the people support a candidate. If we sample 1000 people from the population, and let X be the number of supporters of this candidate among these 1000 people. Let p = X/n be the sample proportion. ^ (1) What is the probability that p > .35? ^ (2) What is the probability that p < .3? ^ 1 ...
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This note was uploaded on 08/24/2009 for the course MATH 262447221 taught by Professor Weisbart during the Spring '09 term at UCLA.
- Spring '09