1.1 Problem Consider a transition matrix summarizing the changes (transitions) between possible states (outcomes). Suppose you have a country divided into three regions: 1, 2 and 3. The entry pij denotes the proportion of the population of region 3' that moves to region 1'. Consider the following transition matrix Consider the following matrix 0.80 0.15 0.05 P: am am am (n am m5nm So for example, 10% of the population of region 1 moves to region 2. Compute the following. Be explicit in the calculations done (not just the results). a. To what number should the columns add? why? b. To what number should the rows add? why? c. Denote the populations of the three regions in millions at some initial point in time 0 by the following 5 : m+m m 6 Where 02 denotes an uncertainty in region 2 due to measurement errors in that census ofce. Find the population of these regions at the beginning of next period d. What is the total population at the beginning of the first period? What was the total population at the initial point? e. Assuming migration patterns remain constant, nd the population of these regions after two periods