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statistics for experimentert

Questions and Answers of Statistics For Experimentert

14. In Prob. 13 let Q = (x - \mu)'A(x - \mu) and show that$$8[(Q - 8(Q))^2] = 8(Q^2) - [8(Q)]^2 = 2 \text{tr} [(AV)^2].$$
13. Let the n x 1 random vector x have a normal density defined by Eq. (10.6.2).Find 8(Q) where$$Q = (x - \mu)'V(x - \mu).$$
12. Find the number K such that the function defined by$$Ke^{-(1/2)Q}$$is a normal density function where Q is defined in Prob. 11.
11. Evaluate the integral$$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(x_1^2-2x_1x_4)e^{-(1/2)Q}dx_1 dx_2 dx_3 dx_4$$where$$Q = 3x_1^2 + 2x_2^2 +
10. In Prob. 9 show that the mean vector of the density of the random vector y is B$\mu$
9. If the k x 1 vector x has a normal density and B is an m x k matrix of rank m, then use Theorem 10.3.1 to show that the density of the m x 1 random vector y has a normal density, where y = Bx.
8. In the normal density given by Eq. (10.6.2) show that $\phi(x)$ is the vector $\mu$ that satisfies $\frac{\partial}{\partial x}N(x; \mu, V) = 0$.
7. Find the constant K such that the following function is a normal density$$f(x_1, x_2) = Ke^{-(2x_1^2+4x_2^2-2x_1x_2-6x_1-4x_2+8)}.$$
6. Define a vector z as z = P'(x - $\mu$), where x is a random n x 1 vector with a normal density given by Eq. (10.6.2) and P is an orthogonal matrix of constants such that P'VP = D, a diagonal
5. Use Theorem 10.5.1 to evaluate $\phi(xy)$ for the random vector with density given in Prob. 1.
4. Use the results of Prob. 3 to find an orthogonal matrix P such that P'RP is a diagonal matrix (see Corollary 10.3.1.1).
3. Find the characteristic roots and characteristic vectors of R in Prob. 1.
2. In Prob. 1 find V and hence show that the covariance of x and y is equal to $ho\sigma_x\sigma_y$.
1. The bivariate normal density can be written as$$N(x, y) = \frac{1}{2\pi\sigma_x\sigma_y\sqrt{1-ho^2}}$$$$\times exp\left[-\frac{1}{2(1-ho^2)}\left[\left(\frac{x-\mu_x}{\sigma_x}\right)^2 -
41. If A is given by$$A = \begin{bmatrix} 2 & 1 & 0 \\\ 1 & 3 & 1 \end{bmatrix}$$find the commutation matrix K23 such that K23 vec(A) = vec(A').
40. If A is an n × n idempotent matrix of rank k, show that [vec(A')]'[vec(A)] = k.
39. If P is an n × n orthogonal matrix, show that [vec(P)]'[vec(P)] = n.
38. Prove Corollary 9.3.6.3.
37. If$$A = \begin{bmatrix} 1 & 0 \\\ 2 & 3 \end{bmatrix}; B = \begin{bmatrix} 1 \\\ 1 \end{bmatrix}.$$find the commutation matrices K22 and K21 such that K22(A × B)K21 = B × A.
36. Prove Corollary 9.2.4.
35. If A and B are given by$$A = \begin{bmatrix} 3 & 2 \\\ 0 & 1 \\\ 1 & -1 \end{bmatrix}; B = \begin{bmatrix} 0 & 1 & -2 \\\ 1 & 3 & 1 \end{bmatrix}.$$show that [vec(A')]'[vec(B)] = tr(AB).
34. Prove Theorem 9.2.1.
33. In Theorem 9.2.2, if A = a' and C = c', where a and c are q x 1 and s x 1 vectors, respectively, and if B is a q X s matrix, show that (a' x c')vec(B) =a'Bc.
32. If A is an m x n matrix with columns ai, show that vec(A) = Σni=1 (ai x ei)= vec[Σni=1 aiei], where ei is the i-th column of the n x n identity matrix.
31. If A is an n x n matrix, show that tr[{vec(A)}{vec(I)}'] = tr(A).
30. Let A and B be m x n matrices. Show that tr (A'B) = tr (AB').
29. If A is an n x n matrix show that tr (A²) ≤ tr (AA').
28. If A is a symmetric n x n matrix and B is an n x n skew-symmetric matrix, show that tr (AB) = 0.
27. If A is an n x n matrix, show that tr (Ak) = 0 for k = 1, 2, 3, ..., if and only if At = 0 for some positive integer t.
26. Let A and B' be m x n matrices such that AB = 0. Show that tr (BCA) = 0 for any m x m matrix C.
25. Let V, A, B be non-negative n x n matrices. Show that AVB = 0 if and only if tr (VAVB) = 0 but that tr (AVB) = 0 does not imply that AVB = 0.
24. Use Prob. 23 to show that A'A = A², if and only if A is symmetric.
23. If A is an n x n matrix and A'A = A², show that tr [(A'A)(AA)] = 0.
22. Let A be an n x n symmetric matrix. Show that A is a positive definite matrix if and only if tr (AB) > 0 for every non-negative matrix B of rank 1.
21. Let A be any n x n matrix of rank k. Show that there exists a nonsingular n x n matrix B such that tr (BA) = k.
20. Let h(x) = Σni=0 aixi be a polynomial. Define h(A) by h(A)= Σni=0 aiAi.where A0 = I and ai are scalars. If λ1, λ2, ..., λn are the characteristic roots of the n x n matrix A, show that tr
19. IfA and A + I are nonsingular 11 x )I matrices, show that
18. Let A be an orthogonal n x n matrix such that det (A + I) ≠ 0. Show that tr [2(A + I)⁻¹ - I] = 0.
17. If A and B are n x n matrices, show that tr [(AB - BA)(AB + BA)] = 0.
16. Let X be an n x p matrix of rank p. Partition X such that X = [X₁, X₂], where X₁ has size n x p₁ and X₂ has size n x p₂ where p₁ + p₂ = p. Show that the rank of B is p₂ where B
15. If A and B are n x n matrices such that AB = 0, show that tr [(A + B)²] = tr (A²) + tr (B²).
14. Let A and B be two n x n matrices such that AB' = 0. Is B'A necessarily equal to zero? Show that tr (B'A) = 0.
13. Let A be an n x n (real) matrix with characteristic roots λ₁, λ₂, ..., λₙ, where anyλᵢ, may not be a real number. Denote λᵢ by xᵢ + iyᵢ, where xᵢ and yᵢ are real numbers
12. If A is defined below, find a 4 x 4 matrix B such that tr (AB) = rank (A).$$A =\begin{bmatrix}3 & 1 & -2 & 0 \\1 & 2 & 3 & -1 \\-2 & 1 & 3 & 4 \\6 & 2 & -2 & -2\end{bmatrix}$$
11. Prove Theorem 9.1.16.
10. Prove Theorem 9.1.14.
9. If A is an n x n symmetric idempotent matrix and V is an nxn positive definite matrix, show that rank (AVA)^-1 = tr(A).
8. If x, is an n x 1 vector for each i = 1, 2, ..., k, and A is an nxn symmetric matrix, show that$$tr \left[A\sum_{i=1}^{k} x_i x_i^T \right] = \sum_{i=1}^{k} x_i^T A x_i.$$
7. Prove Theorem 9.1.13.
6. Prove Theorem 9.1.29.
5. If A, B, and AB are symmetric n x n matrices and the characteristic roots of A are a1, a2,...,a, and of B are b₁, b2,..., bn, show that$$tr (AB) = \sum_{i=1}^{n} a_{j_i} b_i,$$where ajı, aj...,
4. Prove Theorem 9.1.10.
3. Prqve Theorem 9.1.9.
2. Show that tr (al) = na where I is the n x n identity matrix.
1. Prove Theorem 9.1.5.
82. If A is a regular circulant, show that A is also aT-matrix.
81. If A is a regular circulant, show that A' A and AA' are symmetric regular circulants.
79. Prove Corollary 8.10.22.SO. Prove Theorem 8.10.23 by using the fact that P' AP = A for an appropriate permutation (and bence onbogonal) matrix P and Q'BQ .=; B for an appropriate permutation
78. Prove Theorem 8.10.22.
77. Show that if A is a k x k symmetric circulant, then P' AP = A, where P is the permutation matrix in Problem 74.
76. If A is a symmetric circulant, show that A2 is a Tvmatrix and a Csmatrix.
75. In Prob. 74, show that P is a symmetric circulant and that P' = p-l.
74. If A is a k x k regular circulant then B is a k x k symmetric circulant, where P' A = B and hence A = PR, where P = [e .. e*, ek_1> ...• ~, e:zl, where e, is the i-th unit vector (the i-th
73. If A is a *k* x *k* symmetric regular circulant and *k* is an even integer, exhibit the first row of A in terms of *a0*, *a1*, ..., *am*, where *m* = *k*/2.
72. If A and B are *k* x *k* symmetric circulants, show that A2B2 = B2A2, but AB may not equal BA.
71. If A is a symmetric circulant, show that A2 is a symmetric regular circulant.
70. If T is an upper (lower) triangular *n* x *n* matrix and D is a diagonal *n* x *n* matrix, show that DT and TD are upper (lower) triangular matrices.
69. Show that any square matrix A can be written as the sum of a symmetric and a skew-symmetric matrix.
68. If R and T are lower and upper triangular nonsingular matrices, respectively, and if RT = D where D is diagonal, show that R and T are also diagonal.
67. If each entry *pij* of an *n* x *n* correlation matrix R satisfies -1 ≤ *pij* ≤ 1, show that|R| = 0 if and only if at least one *pij* for *i ≠ j* is equal to plus or minus unity.
66. If R is an *n* x *n* correlation matrix, show that |R| attains its maximum value when*pij* = 0 for all *i ≠ j*.
65. Let V be an *n* x *n* covariance matrix and R the corresponding correlation matrix.Show that |V| = *v11* *v22* ... *vnn*.
64. Show that the largest characteristic root of a correlation matrix is less than or equal to *n*, the size of the matrix.
63. Let R be an *n* x *n* correlation matrix and let θ2 be such that θ2 ≤ *pij* for all*i ≠ j*. Show that |R| ≤ 1 - θ2.
62. In the quadratic forms of Eq. (8.8.1), show that each matrix is idempotent and that the product of each pair is equal to the null matrix.
61. Let C be defined by$$C = \begin{bmatrix}2B & -B & -B \\-B & 2B & -B \\-B & -B & 2B\end{bmatrix},$$where B =$$\begin{bmatrix}-1 & -1\end{bmatrix}$$;find the characteristic roots of C.
60. If A is any *k* x *k* matrix, show that there exists a diagonal matrix D where *dii* = +1 or *dii* = -1 such that |A + D| ≠ 0.
59. If either A or B is the null matrix, show that(a) A x B = 0--- OCR End ---
58. If AB = 0 show that(A x F)(B x G) = 0 for any matrices F and G whose sizes are such that multiplication is defined.
57. Find det (A) in Prob. 31.
56. Find the inverse of the matrix B defined by$$B =\begin{bmatrix}4 & 1 & 3 & 2 & 1 \\1 & 2 & 6 & 4 & 2 \\3 & 6 & 12 & 8 & 4 \\2 & 4 & 8 & 24 & 12 \\1 & 2 & 4 & 12 & 12\end{bmatrix}$$Use Theorem
55. Let A be partitioned as in Prob. 50, where A12 = 0 and det (A22) ≠ 0. Find A-1in terms of A11, A21, A22.
54. In Prob. 53, find A-1 and show that B-1A-1 = C-1 where C-1 is defined in Eq. (8.3.2).
53. Find matrices A and B such that AB = C, where C is defined in Eq. (8.3.1), A is lower triangular and does not involve the bi, and B is a diagonal matrix and does not involve the ai.
52. Let the n x n matrix A be defined by A =$$\begin{bmatrix}I_1 & 0 \\B & I_2\end{bmatrix}$$where B is an n1 x n2 matrix and the size of the other submatrices are thus determined. Show that A-1
51. Let A be partitioned as in Prob. 50. If rank (A) = rank (A11), show that A22 =A21A11-1A12.
50. Let A be an n x n matrix that is partitioned as follows:
49. Find the characteristic vectors of the matrix in Prob. 45.
48. Find the characteristic roots of the matrix in Prob. 31.
47. Use Theorem 8.8.10 to find the determinant of the matrix in Example 8.2.1.
46. Use Theorem 8.8.7 to find the inverse of the matrix B in Example 8.2.1. Assume that the inverse exists.
45. Find the characteristic roots of the matrix A where A =$$\begin{bmatrix}2 & 1 & -1 & 0 \\0 & 2 & 1 & -1 \\-1 & 0 & 2 & 1 \\1 & -1 & 0 & 2\end{bmatrix}$$.
44. For the matrices in Prob. 38, demonstrate Theorem 8.8.13.
43. In Prob. 41, find the characteristic roots of A.
42. In Prob. 41, find det (A).
40. In Prob. 39 find det (A x B) in terms of the elements of A and B.41. For the matrix A below, the identity matrices are each 3 x 3. Find the inverse of A.A =$$\begin{bmatrix}31 & 21 \\-1 &
39. Let A be an m x m matrix, and B an n x n upper triangular matrix; show that A x B is an upper triangular block matrix.
38. For the matrices defined below demonstrate (A x B)¹ = A¹ x B¹.A =$$\begin{bmatrix}3 & 2 \\1 & 1\end{bmatrix}$$; B =$$\begin{bmatrix}-6 & 0 \\-1 & 8\end{bmatrix}$$.
37. For the matrices defined below demonstrate that (A x B) (F x G) = (AF) × (BG).A =$$\begin{bmatrix}2 & 1 \\1 & 3\end{bmatrix}$$; B =$$\begin{bmatrix}1 & 3 & -2 \\2 & 0 & -1\end{bmatrix}$$; F
36. For the matrices A and B in Prob. 33 and C defined below, demonstrate that(A x B) x C = A x (B x C).

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