{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# Exam2Spring07 - Math 213 Exam 2(Solutions Prof I.Kapovich...

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 213 Exam 2 (Solutions) Prof. I.Kapovich April 10, 2007 Problem 1 A binary string x of length 20 is chosen at random. Let X be the random variable where X ( x ) is the number of 0s in x plus twice the number of 1s in x . Find E ( X ) and V ( X ). Solution. For i = 1 , 2 , . . . , 20 let X i = 1 , if the i-th bit is 0 2 , if the i-bit is 1 Then X = X 1 + · · · + X 20 . Clearly, the random variables X 1 , . . . , X 20 are independent. Therefore EX = ∑ 20 i =1 EX i = 20 EX 1 and V X = ∑ 20 i =1 V X i = 20 V X 1 . We have EX 1 = 1 · 1 2 + 2 · 1 2 = 3 2 and E ( X 2 1 ) = 1 2 · 1 2 + 2 2 · 1 2 = 5 2 Therefore EX = 20 X i =1 EX i = 20 EX 1 = 20 · 3 2 = 30 , and V ( X ) = 20 V X 1 = 20 ( E ( X 2 1 )- ( EX 1 ) 2 ) = 20 5 2- 9 4 = 20 · 1 4 = 5 . Problem 2 Find the general solution of the recurrence relation ( † ) a n = 4 a n- 1- 4 a n- 2 + n. Solution. The associated homogeneous relation is a n = 4 a n- 1- 4 a n- 2 . It has characteristic equation r 2 = 4 r- 4, that is r 2- 4 r...
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

Exam2Spring07 - Math 213 Exam 2(Solutions Prof I.Kapovich...

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

View Full Document
Ask a homework question - tutors are online