
13. (4 points) Suppose that A satisfies (a) A has two eigenvalues r and r+1, (0)...
(n)," 2 0) be the two-state Markov chain on states (. i} with transition probability matrix 0.7 0.3 0.4 0.6 Find P(X(2) 0 and X(5) X() 0)
0.5 0 0 5. Let P 0.5 0.6 0.3represent the probability transition matrix of a Markov chain with three 0 0.4 0.7 states (a) Show that the characteristic polynomial of P is given by P-ÀI -X-1.8λ2 +0.95λ-0.15) (b) Verify that λι 1, λ2 = 0.5 and λ3 = 0.3 satisfy the characteristic equation P-λ1-0 (and hence they are the eigenvalues of P) c) Show thatu3u2and u3are three eigenvectors corresponding to the eigenvalues λι, λ2 and λ3, respectively 1/3 (d) Let...
A Markov chain {Xn, n ≥ 0} with state space S = {0, 1, 2} has transition probability matrix 0.1 0.3 0.6 p = 0.5 0.2 0.3 0.4 0.2 0.4 If P(X0 = 0) = P(X0 = 1) = 0.4 and P(X0 = 2) = 0.2, find the distribution of X2 and evaluate P[X2 < X4].
4. RWI 4.5.13). Suppose that the sequence (xn) satisfies n 1,2,.. X2ax.1 + bx and that 0< r<R satisfy - az- b (z-rz- R) (a) Show that x O(R" ). O(2") is false. (b) Give an example with r R 2 for which x, (c) What asymptotic (big-oh) estimate for (x) can you give in general if r R> 0?
8th Problem [4 points] C(s) stp Assume the closed loop system which is excited by a ramp r(t)=t1(t) Step Response 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 2 3 4 5 7 Time (seconds) [4 p] Let pE 1,10and the desired response be . Select H(s) and C(s). Justify your selection epndun
(23). (15 Marks). Suppose A has eigenvalues 11 = 3, 12 = 1, 13 = 0) with corresponding eigenvectors 0 0 2 = [1] 12 8) 23 = (a) (5 points). How do you know that the third column of A contains all zeros? (b) (10 points). Find the matrix A.
The accompanying tree diagram represents a two-stage experiment. (Let x = 0.3, y = 0.7, r = 0.6, s = 0.4, t = 0.5, and w = 0.5.) Label all branches of the tree diagram and final outcomes. (Note: final outcomes are the results of the Product Rule). Use the diagram to find the following. Provide exact results: 1. ?(?) 2. ?(? ? ) 3. ?(?| ?) 4. ?(?| ?) 5. ?(? ? ∪ ?) 6. ?(? ∪ ?)
Problem 4 (4 points each). Let S = R {0}. (a) Let f: S R be f(x) = cos(1/x). Show that lim-0 f(x) does not exist. (b) For any fixed a > 0, let f: S+R be f(x) = rºcos(1/x). Show that lim -- f(x) = 0. (c) Find a value be R for which the function f: R+R given by f(x) = { 2" cos(1/x) if r +0, if x = 0, is continuous at 0. Is this b...
The diode in the circuit below has a saturation current Is-10-13 A and n=1. Its ID-VD curve is illustrated below for your convenience. Problem 2 a) Using the graphical method, determine the value of R that would result in a diode current Ip -4mA. What is the resulting voltage Vp? b) With R as calculated above, a sinusoidal signal v, 100 sin(cot) mV is superimposed on the DC source. Draw the small signal model and find the output voltage across...
2. [-12 Points) DETAILS LARLINALG8 7.2.005. Consider the following. -4 20 0 1 -3 A = 040 P= 04 0 4 0 2 1 2 2 (a) Verify that A is diagonalizable by computing p-AP. p-1AP = 11 (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n x n matrices, then they have the same eigenvalues. (91, 12, 13)...