title: Markov Chains and equilibrium probability
body: <p>The fire danger during the summer in Mount Baker National Forest is classified into one of three danger levels. These are 1 =low, 2 =moderate, 3 =high. The probability of daily transitions between these states is given by the following flow diagram:</p>
  
  <p>(a) Write the model in matrix form to project the fire danger probability from one day to the next.</p>
  
  <p>Image: <a href="http://i.stack.imgur.com/TRuUy.png" rel="nofollow">http://i.stack.imgur.com/TRuUy.png</a></p>
  
  <p>(b) If we are in State 1 today, what is the probability that we will be in State 2 the day after tomorrow?</p>
  
  <p>(c) If the matrix you found is correct, then it has eigenvalues and eigenvectors given by</p>
  
  <pre><code>Lambda = [
  1.0000     0         0
  0          0.0697    0
  0          0         0.4203
  
  R=
  -0.4699   -0.5551    -0.7801
  -0.7832   0.7961     0.1813
  -0.4072   -0.2410    0.5988   
  </code></pre>
  
  <p>Based on these, what is the equilibrium probability of being in each state?</p>
  
  <p>I found the matrix form for part a, I could not figure out part b and c. Thank you</p>
  
  <pre><code>  A = [0.5    0.3    0
       0.4    0.5    0.5
       0.1    0.2    0.5]
  </code></pre>