hw8sol - FALL 2011 ORIE 3500/5500 PROBLEM SET 8 SOLUTIONS...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
FALL 2011 ORIE 3500/5500 PROBLEM SET 8 SOLUTIONS (1) (a) By Chebyshev’s inequality, P [ | X n - p | > 0 . 2] Var( X n ) 0 . 2 2 = p (1 - p ) 0 . 04 n . We choose n large enough that this value is less than 0.1 for any p . Since p (1 - p ) 1 / 4 for 0 p 1 (plot the function g ( p ) = p - p 2 ), we require 0 . 16 n 0 . 1 - 1 , i.e. n 62 . 5. Therefore, we should interview at least 63 people. (d) We seek n such that P [ X n > 0 . 5] 0 . 9. Since one way for X n > 0 . 5 to be true is that X n lies within 0.1 of its mean p = 0 . 6, we have P [ X n > 0 . 5] P [ | X n - 0 . 6 | ≤ 0 . 1] 1 - 0 . 6(0 . 4) 0 . 01 n . Hence, it is enough to choose n to make this at least 0.9, i.e. 0 . 24 / (0 . 01 n ) 0 . 1. This is satisfied for n 240. Therefore, when p = 0 . 6, the probability that we predict correctly will be at least 0.9 provided we interview at least 240 people. Alternatively, observing that n X n = n i =1 X i Bin( n, 0 . 6), we could compute this probability exactly for different values of n , and stop as soon as we exceed 0.9.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 3

hw8sol - FALL 2011 ORIE 3500/5500 PROBLEM SET 8 SOLUTIONS...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online