hw505-07-1-solution

# hw505-07-1-solution - IE 505 SKETCH OF SOLUTIONS TO...

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

IE 505 SKETCH OF SOLUTIONS TO HOMEWORK # 1 1. Mass function of X 1 conditioned on X 1 + X 2 = n is simply Pr { X 1 = i | X 1 + X 2 = n } for i = 0 , 1 ,...,n : Pr { X 1 = i | X 1 + X 2 = n } = Pr { X 1 = i,X 1 + X 2 = n } Pr { X 1 + X 2 = n } = Pr { X 1 = i,X 2 = n - i } Pr { X 1 + X 2 = n } = e - λ 1 λ i 1 e - λ 2 λ n - i 2 i !( n - i )! n ! e - ( λ 1 + λ 2 ) ( λ 1 + λ 2 ) n = ˆ n i ! ( λ 1 / ( λ 1 + λ 2 )) i ( λ 2 / ( λ 1 + λ 2 )) n - i . Note that the resulting mass function is Binomial. 2. Pr { N = n } = Pr { X i a for i n - 1 and X n > a } = [ F ( a )] n - 1 (1 - F ( a )). 3. In general, let F ( x ) be the distribution function for the location of the ﬁre. The problem is to minimize G ( a ) = E [ | X - a | ], where a is the location of the station. G ( a ) = Z a 0 ( a - x ) dF ( x ) + Z a ( x - a ) dF ( x ) . It is easy to verify that G ( a ) is convex by checking G 00 ( a ) = 2 dF ( a ) 0. Hence, mini- mizing a can be found by G 0 ( a ) = R a 0 dF ( x ) + R a dF ( x

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.

## This note was uploaded on 11/02/2009 for the course IE 505 taught by Professor L during the Spring '09 term at Bilkent University.

### Page1 / 2

hw505-07-1-solution - IE 505 SKETCH OF SOLUTIONS TO...

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

View Full Document
Ask a homework question - tutors are online