- 9.9.33E: In exercise 23 of Section 9.8, let Ck be the event that the gambler...
- 9.9.1E: Suppose P(A | B) = 1/2 and P(A ? B) = 1/6. What is P(B)?
- 9.9.2E: Suppose P(X | Y ) = 1/3 and P(Y ) = 1/4. What is P(X ? Y )?
- 9.9.3E: The instructor of a discrete mathematics class gave two tests. Twen...
- 9.9.4E: a. Prove that if A and B are any events in a sample space S, with P...
- 9.9.5E: Suppose that A and B are events in a sample space S and that P(A), ...
- 9.9.6E: An urn contains 25 red balls and 15 blue balls. Two are chosen at r...
- 9.9.8E: A pool of 10 semifinalists for a job consists of 7 men and 3 women....
- 9.9.9E: Prove Bayes’ Theorem for n = 2. That is, prove that if a sample spa...
- 9.9.10E: Prove the full version of Bayes’ Theorem.
- 9.9.11E: One urn contains 12 blue balls and 7 white balls, and a second urn ...
- 9.9.12E: Redo exercise 11 assuming that the first urn contains 4 blue balls ...
- 9.9.13E: One urn contains 10 red balls and 25 green balls, and a second urn ...
- 9.9.14E: A drug-screening test is used in a large population of people of wh...
- 9.9.15E: Two different factories both produce a certain automobile part. The...
- 9.9.16E: Three different suppliers—X, Y , and Z—provide produce for a grocer...
- 9.9.17E: Prove that if A and B are events in a sample space S with the prope...
- 9.9.18E: Prove that if P(A ? B) = P(A) · P(B), P(A) ? 0, and P(B) ? 0, then ...
- 9.9.19E: A pair of fair dice, one blue and the other gray, are rolled. Let A...
- 9.9.20E: Suppose a fair coin is tossed three times. Let A be the event that ...
- 9.9.21E: If A and B are events in a sample space S and A ? B = ?, what must ...
- 9.9.22E: Prove that if A and B are independent events in a sample space S, t...
- 9.9.23E: A student taking a multiple-choice exam does not know the answers t...
- 9.9.24E: A company uses two proofreaders X and Y to check a certain manuscri...
- 9.9.25E: A coin is loaded so that the probability of heads is 0.7 and the pr...
- 9.9.27E: The example used to introduce conditional probability described a f...
- 9.9.28E: A coin is loaded so that the probability of heads is 0.7 and the pr...
- 9.9.29E: Suppose that ten items are chosen at random from a large batch deli...
- 9.9.30E: Suppose the probability of a false positive result on a mammogram i...
- 9.9.31E: Empirical data indicate that approximately 103 out of every 200 chi...
- 9.9.32E: In exercise 23 of Section 9.8, let ?Ck ?be the event that the gambl...
Solutions for Chapter 9.9: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications | 4th Edition
Tv = Av + Vo = linear transformation plus shift.
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
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).
= Xl (column 1) + ... + xn(column n) = combination of columns.
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 •
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.