Hw4_ORIE361_2008_sol - ORIE 361/523 Homework 4 Solutions Instructor Mark E Lewis 1 For large n we will estimate P(Xn = 1|X0 = 1 by the steady state

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

View Full Document Right Arrow Icon
ORIE 361/523 – Homework 4 Solutions Instructor: Mark E. Lewis February 26, 2008 1. For large n we will estimate P ( X n = 1 | X 0 = 1) by the steady state probability of state 1. Suppose the steady state probability or the stationary distribution is given by π = ( π 1 2 3 ). Then we know that π satisfies π = πP and π 1 + π 2 + π 3 = 1 . To solve these equations is same solving the system 0 = b where A = - 1 0 2 / 5 1 - 3 / 4 3 / 5 1 1 1 and b = 0 0 1 Solving this we get π = (0 . 1463 , 0 . 4878 , 0 . 3659). So the answer for our problem is 0 . 1463 2. Suppose X n is a Markov chain modeling this and X n = 1 or 2 depending on whether dividend has been paid or not. We need to solve for the stationary distribution of the stochastic matrix P = ± 0 . 9 0 . 1 0 . 6 0 . 4 ² The stationary distribution is given by (6 / 7 , 1 / 7). That means, dividend will be paid every 6 times out of 7 on an average in the long run. 3. Let X n be 1, 2 or 3 depending on whether the n th day is sunny, cloudy or rainy respectively. Then X n is a Markov chain with transition probability matrix P = 0 1 / 2 1 / 2 1 / 4 1 / 2 1 / 4 1 / 4 1 / 4 1 / 2 To get the stationary distribution we need to solve for π = ( π 1 2 3 ) from the equations
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.

This note was uploaded on 03/01/2010 for the course ORIE 361 taught by Professor Lewis,m. during the Spring '07 term at Cornell University (Engineering School).

Page1 / 3

Hw4_ORIE361_2008_sol - ORIE 361/523 Homework 4 Solutions Instructor Mark E Lewis 1 For large n we will estimate P(Xn = 1|X0 = 1 by the steady state

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