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- 184.108.40.206.119: (Rutherford-Geiger experiments) In 1910. E. Rutherford and H. Geige...
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- 22.214.171.124.122: Let P = lo/c be the probability that a certain type of lightbulb wi...
- 126.96.36.199.123: Guess how much less the probability in Prob. 10 would be if the sig...
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- 184.108.40.206.125: Suppose thar a test for extrasensory perception consists of naming ...
- 220.127.116.11.126: A carton contains 20 fuses, 5 of which are defective. Find the prob...
- 18.104.22.168.127: (Multinomial distribution) Suppose a trial can result in precisely ...
- 22.214.171.124.128: TEAM PROJECT. Moment Generating Function. The moment generating fun...
Solutions for Chapter 24.7: Binomial. Poisson, and Hypergeometric Distributions
Full solutions for Advanced Engineering Mathematics | 9th Edition
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.
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).
= Xl (column 1) + ... + xn(column n) = combination of columns.
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.
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 II·
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.