- 126.96.36.199.96: If 100 flips of a coin result in 30 heads and 70 tails. can we asse...
- 188.8.131.52.97: If in 10 flips of a coin we get the same ratio as in Prob. I (3 hea...
- 184.108.40.206.98: What would be the smallest number of heads in Prob. I under which t...
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- 18.104.22.168.100: Solve Prob. 4 if the sample is 25, 31. 33, 27, 29. 35.
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- 18.104.22.168.106: Verify the calculations in Example 1 of the text.
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- 126.96.36.199.108: Using the given sample, test that the corresponding population has ...
- 188.8.131.52.109: Can we assert that the traffic on the three lanes of an expressway ...
- 184.108.40.206.110: If it i5 known that 25% of certain steel rod~ produced by a standar...
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- 126.96.36.199.114: CAS EXPERIMENT. Random Number Generator. Check your generator expel...
- 188.8.131.52.115: TEAM PROJECT. Difficulty with Random Selection. 77 students were as...
Solutions for Chapter 25.7: Goodness of Fit. x2-Test
Full solutions for Advanced Engineering Mathematics | 9th Edition
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
A sequence of steps intended to approach the desired solution.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Every v in V is orthogonal to every w in W.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Constant down each diagonal = time-invariant (shift-invariant) filter.