hw3sol - Computer Science 340 Reasoning about Computation...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Computer Science 340 Reasoning about Computation Homework 3 Due at the beginning of class on Wednesday, October 10, 2007 Problem 1 Suppose you toss three fair, mutually independent coins. Define the following events: A - the event that the first coin is heads B - the event that the second is heads C - the event that the third coin is heads D - the event that the number of heads is even Determine the maximum value of k such that these events are k -wise independent. Solution: These events are 3-wise independent. First, let us note that these events are not mutually independent and thus they are not 4-wise independent. Indeed, Pr( A B C D ) = 0 . Here, we used the fact that if the events A , B and C happen, then the number of heads is odd, so the event D does not happen. On the other hand Pr( A ) · Pr( B ) · Pr( C ) · Pr( D ) = 1 16 , thus Pr( A B C D ) 6 = Pr( A ) · Pr( B ) · Pr( C ) · Pr( D ) . Now we prove that any three of the events A , B , C and D are mutually independent. The events A , B , C are mutually independent by the condition. Let us prove that A , B , and D are mutually independent. Let I A , I B , and I D be the indicators of the events A , B and D . Fix arbitrary x 1 , x 2 , x 3 ∈ { 0 , 1 } . Let us compute the following probabilities: Pr( I A = x 1 ) = 1 2 ; Pr( I B = x 2 ) = 1 2 ; Pr( I D = x 3 ) = 1 2 ;
Background image of page 1

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

View Full DocumentRight Arrow Icon
Pr( I A = x 1 I B = x 2 I D = x 3 ) = Pr( I A = x 1 I B = x 2 I A I B I C = x 3 ) = Pr( I A = x 1 I B = x 2 x 1 x 2 I
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This homework help was uploaded on 01/29/2008 for the course COS 340 taught by Professor Charikarandchazelle during the Fall '07 term at Princeton.

Page1 / 5

hw3sol - Computer Science 340 Reasoning about Computation...

This preview shows document pages 1 - 3. Sign up to view the full document.

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