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# Solutions for Chapter 6.3: Discrete Probability

## Full solutions for Discrete Mathematics and Its Applications | 6th Edition

ISBN: 9780073229720

Solutions for Chapter 6.3: Discrete Probability

Solutions for Chapter 6.3
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##### ISBN: 9780073229720

This expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.3: Discrete Probability includes 23 full step-by-step solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 6. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073229720. Since 23 problems in chapter 6.3: Discrete Probability have been answered, more than 36144 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

• Column space C (A) =

space of all combinations of the columns of A.

• Condition number

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 IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

• Diagonalizable matrix A.

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

• Elimination.

A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

• Fibonacci numbers

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

• 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.

• Hermitian matrix A H = AT = A.

Complex analog a j i = aU of a symmetric matrix.

• Incidence matrix of a directed graph.

The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

• Iterative method.

A sequence of steps intended to approach the desired solution.

• Kirchhoff's Laws.

Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

• Left inverse A+.

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

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• Plane (or hyperplane) in Rn.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

• 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.

• Sum V + W of subs paces.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

• Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

T- 1 has rank 1 above and below diagonal.

• Volume of box.

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

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