 12.5.12.1.240: A list of all possible outcomes of an experiment is called a(n) space.
 12.5.12.1.241: Each individual outcome in a sample space is called a sample .
 12.5.12.1.242: If a first experiment can be performed in two distinct ways and a s...
 12.5.12.1.243: A helpful method to determine a sample space is to construct a(n) d...
 12.5.12.1.244: Selecting States If two states are selected at random from the 50 U...
 12.5.12.1.245: Selecting Dates If two dates are selected at random from the 365 da...
 12.5.12.1.246: Selecting Golfballs A bag contains six golfballs, all the same size...
 12.5.12.1.247: Remote Control Your television remote control has buttons for digit...
 12.5.12.1.248: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.249: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.250: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.251: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.252: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.253: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.254: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.255: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.256: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.257: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.258: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.259: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.260: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.261: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.262: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.263: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.264: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.265: In Exercises 926, use the counting principle to determine the answe...
 12.5.12.1.266: Three Chips Suppose that a bag contains one white chip and two red ...
 12.5.12.1.267: Thumbtacks A thumbtack is dropped on a concrete floor. Assume that ...
 12.5.12.1.268: Ties All my ties are red except two. All my ties are blue except tw...
 12.5.12.1.269: Rock Faces An experiment consists of 3 parts: flipping a coin, toss...
 12.5.12.1.270: Jumble For Exercises 3132, refer to the Recreational Math box on pa...
 12.5.12.1.271: Jumble For Exercises 3132, refer to the Recreational Math box on pa...
Solutions for Chapter 12.5: Probability
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 12.5: Probability
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 32 problems in chapter 12.5: Probability have been answered, more than 74082 students have viewed full stepbystep solutions from this chapter. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9. Chapter 12.5: Probability includes 32 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Column space C (A) =
space of all combinations of the columns of A.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Multiplier eij.
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.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.