Chapter 8 PreSkills 1 1 1 8 1] Pivot around element A22 given that 6 2 5 0 2 0 3 2 Z] A probability distribution must have columns that sum to one, Consider the event of tossing a coin twice in a row which has a sample space that has four elements. Event Probablllty Tossing no heads Tossing one head Tossing exactly 2 heads 3] Write the given matrix as a system of linear equations. [10 21 130 4] Multiply the matrices. 1 1 [1 1] 0113 5] Find the inverse of the given matrix. .7 .8 .1.4 Chapter 8 Markov Processes In this area, we are trying to predict the future. There is a simple type of dependency that occurs frequently in applications and can be analyzed with ease. We suppose that the probabilities of the Various outcomes of the current experiment depend (at most) on the outcome of the preceding experiment. The sequence of experiments is called a Markov process. First let's talk about the transition matrix of the Markov process. This is where we keep track of the change from the current state to the next state. The matrix we create is called a STOCHASTIC matrix. This word comes from the Greek word Stochastices which means \"a person who predicts the future." Ex 1: Census studies from the 19605 reveal that, in the U.S., 80 % of the daughters of women in the labor force also worked outside of the home but 70% of daughters of women not in the labor force also were not in the labor force. Current State Work Don't N S Work E T Work X A T T Don't E Work Ex 2: A particular utility stock is very stable, and in the short run, the probability that it increases or decreases in price depends only on the result of the preceding day's trading. Suppose that if the stock increases one day, the probability that on the next day it increases is .3 and decreases is .5. 0n the other hand, if the stock is unchanged one day, the probability that on the next day it increases is .6 and .1 that it remains unchanged. If the stock decreases one day, the probability that is remains unchanged the next day is .4 and decreases is .3. Form the transition matrix. Current State NS ET XA TT Matrices MUST meet two conditions to be Stochastic: 1) All entries are greater than or equal to 0 (i .e. NON -negative entries) 2) The sum of the entries in each column is 1. Now, we will focus on how we can use the transition matrix to predict the FUTURE... (insert cool music here) Return to the first example. Assume that the trend remains unchanged from one generation to the next. In 1960. about 40% of US. women worked outside the home. This is the distribution matrix at generation zero, We use a column matrix to describe this information. [' 2] Be sure the order matches the transition: work then no work. To determine how many women will work outside of the home for the next generation. we use Transition times Distribution (Gen 0) which equals Distribution (Gen 1) This can we repeated for each generation. This leads to a formula: A"[ lo=[ 1n where n is the number of generations after the ' tial distribution (the current state). Example 1 Continued: Thus of the women will work after one generation and of the American women will be working after two generations. Example 2 Continued: Determine the state of the utilities after 2 days, if the initial distribution shows .2 increases, .3 unchanged. and .5 decreases. Practice... 1) The Taxi Problem Taxis pick up and deliver passengers in a city that is divided into three zones. Records kept by the drivers show that, of the passengers picked up in Zone 1 50% are taken to a destination in zone I. 40% are taken to zone 1], and 10% to zone 111. Refer to the transition diagram for the rest. Suppose that at the beginning of the day, 60% of the taxis are in zone I and 10% are in zone 11. a) Create a matrix that shows the transition. a matrix that shows the initial distribution. and determine the distribution of taxis in the various zones after all have had one passenger. h) Then nd the distribution after 2 passengers. The TAXI problem Part 2: (Using the same diagram for part1) Write a transition matrix then determine the distribution of taxis in the various zones after all have had one passenger given that at the beginning of the day 80% of the taxis are in zone 1 and 10% are in zone 3. Find the distribution of taxis after 1 passenger. Page 5 of 6 - ZOOM + 2) Write a transition matrix with proper labeling. 0.075 0.9 Bull Bear market market 0.8 0.15 0.025 0.25 0.25 0.05 Stagnant market 05 3) Write a transition matrix with proper labeling. 0.4 Sunny Partly 0.4 cloudy Rainy 0.64) The MOUSE Mice in a T-maze have the choice of turning left and are rewarded with cheese or turning right to receive cheese with a mild shock. After the first day, meir decision is inuenced by what happened on the previous day. Mouse Of those that go left on a certain day, 90% go to the left on the next day. Of those that go right on a certain day, 30 % will go to the right the next day. What percentage of mice will be going to the left after many days (nding a long term trend means tojust do many generations until it settles into a percentage)? a) Write a transition matrix based on the information given. b) If the initial distribution shows that of the mice that go right on a certain day 70% will go left the next day and 30% go to the right, find the first generation distribution of mice. 0) Find an approximation for the long term trends