EXERCISE 3.1. Let be an irreducible transition matrix on X, and let be a probability distribution on X. Show that the transition matrix P(x,y) V(x,y) x(y) (y.x) x(x)V(x,y) 1- ^ if y + x, if y = z (=)V(=,x) Σ Ψ(1,2) x(x)V (1,2) 2:27x defines a reversible Markov chain with stationary distribution. ^
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- Let B = {-3,2), (4, -2)} and B' = {(-1,2), (2,-2)} a. Find the transition matrix from B to B'. b. Find the coordinate matrix [x]B given that [x]B -G]-Let B = {ü,,ü,}, B'={v,v,} be two bases in R, where %| ü, %3D 3 (a) Find the transition matrix from B' to B. (b) Find the transition matrix from B to B'.Find the transition matrix from B to B'. B = {(-1, 0, 0), (0, 1, 0), (0, 0, –1)}, B' = {(0, 0, 6), (1, 4, 0), (8, 0, 6)} %3D
- 4. Let X'= (X₁,..., Xn) be an n-dimensional random vector whose covariance matrix exists. Let A be an m x n matrix of constants. Then Cov(AX) = ACov(X)A'. True or false, give reasoning.Consider the following. B = {(1, 2, 16), (-1, 1, 8), (-3, 3, 28)), B' = {(10, 3, 6), (3, 1, 2), (6, 2, 5)}, B 2 (a) Find the transition matrix from B to B'. [x] = p-1 = (b) Find the transition matrix from B' to B. P = (c) Verify that the two transition matrices are inverses of each other. pp-1 = ↓↑ [x]B ↓↑ ↑ (d) Find the coordinate matrix [x]B, given the coordinate matrix [x]g'.3 Let B be the OLS estimate from the regression of y on X. Let A be a (k + 1) × (k + 1) nonsingular matrix and define z, = x,A, t = 1, ..., n. Therefore, z, is 1 × (k + 1) and is a nonsingular linear com- bination of x,. Let Z be then x (k + 1) matrix with rows z,. Let B denote the OLS estimate from a regression of y on Z. (i) Show that B = A-'B. (ii) Let ŷ, be the fitted values from the original regression and let ŷ, be the fitted values from regress- ing y on Z. Show that y, sions compare? (iii) Show that the estimated variance matrix for ß is ô'A-(X'X)-'A-", where &² is the usual vari- ance estimate from regressing y on X. (iv) Let the B; be the OLS estimates from regressing y, on 1, x,1, ..., Xk, and let the B; be the OLS es- timates from the regression of y, on 1, a,x,1,..., ax, where a; + 0, j = 1, ..., k. Use the results from part (i) to find the relationship between the B; and the B;. (v) î, for all t = 1, 2, .., n. How do the residuals from the two regres- = se(B,)/la,l.…
- b b M "O © 56% 14:13 webwork.yeditepe.edu.tr 26 (a) Let -2 S = V1 = V2 = -5 13 Find vectors = In U2 = in R? such that S is the transition matrix from {v1, v2} to {u1, u2}. (b) Let P, be the vectors space of all polynomials of degree less than four. Find the transition matrix A representing the change of basis from the ordered basis {-2, — 4г, — 12а?, 1523} to {1,1+ x, 1+ x + x²,1+ x+ x² + x³ }. A = İngilizce TürkçeConsider the following. B = {(3, 8, 4), (1, 4, 2), (2, 8, 5)}, B' = {(10, 3, 3), (3, 1, 1), (-6, -2, -1)}, 1 [x]g = (a) Find the transition matrix from B to B'. p-1 = (b) Find the transition matrix from B' to B. P = (c) Verify that the two transition matrices are inverses of each other. Pp-1 P Type here to search 1O 67°F 近 1 1Example 18.2 For the probability density of a system of random variables (X, Y): f(x, y) = 0.5 sin (x + y) (0 < x < 7,0 << 7), 2' determine (a) the distribution function of the system, (b) the expectations of X and Y, (c) the covariance matrix.
- Let X be a 4-dimensional random vector defined as X = correlation matrix E[X] = 0 0 0 0 Let Y be a 3-dimensional random vector with Rx = [X₁ X2 X3 X4]' with expected value vector and 1 0 0 Y₁ = X₁ - X2, Y₂=X2- X3, Y3 = X3 X4. -1 7 1 -1 0 0 -1 1 (a) Find a matrix A such that Y = AX. (b) Find the correlation matrix of Y, that is Ry. (c) Find the correlation matrix between X₁ and Y, that is Rx₁Y. 0 0 -1 1Consider the two Dases S 3 {(1,0, 0), (0, 1,0), (0,0, 1)} and в — {(1,0, 1), (0, -1,2), (2, 3, —5)} for R' (Hint: Throughout this problem, refer to section 5 in the notes on Canvas, or the relevant sections in your textbook). (a) Find the transition matrix P 1 from the standard basis S to the nonstandard basis B. (b) Use your answer from part (a) to find the coordinate vector of the vector x = (1,2, –1) relative to the basis B. (c) Find the transition matrix from B to S. (d) Use your answer from part (c) to express the vector x = standard basis S. Give the coordinates for this vector relative to the basis S. (5, 1, 3) in terms of theConsider the following. B = {(-1, 4, 1), (-2, 4, 1), (-2, 4, 2)}, B' = {(10, 3, -3), (3, 1, -1), (6, 2, -1)}, [x]B' = (a) Find the transition matrix from B to B', p-1- (b) Find the transition matrix from B' to B. P= (c) Verify that the two transition matrices are inverses of each other. pp-1. (d) Find the coordinate matrix [x]B, given the coordinate matrix [x]8. [x]B =