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# Solutions for Chapter 24.7: Binomial. Poisson, and Hypergeometric Distributions

## Full solutions for Advanced Engineering Mathematics | 9th Edition

ISBN: 9780471488859

Solutions for Chapter 24.7: Binomial. Poisson, and Hypergeometric Distributions

Solutions for Chapter 24.7
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##### ISBN: 9780471488859

Chapter 24.7: Binomial. Poisson, and Hypergeometric Distributions includes 16 full step-by-step solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Since 16 problems in chapter 24.7: Binomial. Poisson, and Hypergeometric Distributions have been answered, more than 46198 students have viewed full step-by-step solutions from this chapter. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This expansive textbook survival guide covers the following chapters and their solutions.

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

Parentheses can be removed to leave ABC.

• Change of basis matrix M.

The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

• 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.

• Exponential eAt = I + At + (At)2 12! + ...

has derivative AeAt; eAt u(O) solves u' = Au.

• Graph G.

Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

• Hypercube matrix pl.

Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

• lA-II = l/lAI and IATI = IAI.

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

• Left inverse A+.

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

• 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).

• Multiplication Ax

= Xl (column 1) + ... + xn(column n) = combination of columns.

• Network.

A directed graph that has constants Cl, ... , Cm associated with the edges.

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• 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.

• Plane (or hyperplane) in Rn.

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

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• 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.

• Triangle inequality II u + v II < II u II + II v II.

For matrix norms II A + B II < II A II + II B IIĀ·

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

T- 1 has rank 1 above and below diagonal.

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