Let A be an n n positive stochastic matrix with dominant eigenvalue 1 = 1 and linearly independent eigenvectors x1, x2, ... , xn, and let y0 be an initial probability vector for a Markov chain y0, y1 = Ay0, y2 = Ay1, ... (a) Show that 1 = 1 has a positive eigenvector x1. (b) Show that yj1 = 1, j = 0, 1, .... (c) Show that if y0 = c1x1 + c2x2 ++ cnxn then the component c1 in the direction of the positive eigenvector x1 must be nonzero. (d) Show that the state vectors yj of the Markov chain converge to a steady-state vector. (e) Show that c1 = 1 x11 and hence the steady-state vector is independent of the initial probability vector y0.

Elementary Statistics Math 124-08 Vera Klimkovsky 21 September 2016 Chapter 4 – Scatter plots and Correlation Chapter 4 – Scatter plots and Correlation Scatter plot ( XY plot) 3 ways to describe a scatter plot: o Form: Linear or Nonlinear o Direction: Positive or Negative ( could also be neither) o Strength: Strong, Moderate, Weak Ex: Two Quantitative...