2. [5 points each] The birth and death process {X(t), t 2 0} on non-negative integers with parameters An = 0, / = (2 0), n >0 and / = 0 is called a pure death process. (a) Sketch the sample paths of this pure death process with X(0) = 1. (b) Explain briefly how to simulate this pure death process. (c) When ( = 1, find E[X(() X(0) = 1].4. (15 points) Consider the linear system 2x1 - 22+23 =-1 21 + 2x2 + 2x3 =4 (1) -21 - 22 + 2x3 = -5 (a) show that p(B) = v5 in the Jacobi method. (b) Does the Jacobi method converge?Consider the following system of equations 4x + y - z =13 2x - 10y - 2z = -16 4x - 2y - 12z= - 4 a) Will the system converge to a solution using Jacobi Iteration? Why? (5 pts) b) Use Jacobi iteration to find starting with . (15 pts)Exercise Write down the iterative schemes for the Jacobi, Gauss-Seidel and SOR methods. Explain how SOR is obtained from the Gauss-Seidel method. Explore convergence property of the Jacobi and SOR method for the system Anr = b 2 -1 0 -1 2 . . . An = 0 b = [1. .. 1] n = 30 2 0 0 2 Use r(0) = [000. . . 0], Wopt = 1+ sin Iterate until ||x - x*)||