 24.2.24.1.21: Graph a sample space for the expeliment: Tossing 2 coins
 24.2.24.1.22: Graph a sample space for the expeliment: Drawing 4 screws from a lo...
 24.2.24.1.23: Graph a sample space for the expeliment: Rolling 2 dice
 24.2.24.1.24: Graph a sample space for the expeliment:Tossing a coin until the fI...
 24.2.24.1.25: Graph a sample space for the expeliment:Rolling a die until the fir...
 24.2.24.1.26: Graph a sample space for the expeliment:Drawing bolts from a lot of...
 24.2.24.1.27: Graph a sample space for the expeliment:Recording the lifetime of e...
 24.2.24.1.28: Graph a sample space for the expeliment:Choosing a committee of 3 f...
 24.2.24.1.29: Graph a sample space for the expeliment:Recording the daily maximum...
 24.2.24.1.30: In Prob. 3, circle and mark the events A: Equal faces, B: Sum excee...
 24.2.24.1.31: In rolling 2 dice, are the events A: SIIII1 divisible by 3 and B: S...
 24.2.24.1.32: Answer the question in Prob. 11 for rolling 3 dice.
 24.2.24.1.33: In Prob. 5 list the outcomes that make up the event E: First "Six" ...
 24.2.24.1.34: List all 8 subsets of the sample space S = (a, b, c}.
 24.2.24.1.35: In connection with a trip to Europe by some students, consider the ...
 24.2.24.1.36: Using Venn diagrams, graph and check the mles A U (B n C) = (A U B)...
 24.2.24.1.37: (De Morgan's laws) Using Venn diagrams. graph and check De Morgan's...
 24.2.24.1.38: Using a Venn diagram. show that A <;;; B if and only if An B = A.
 24.2.24.1.39: Show that, by the definition of complement, for any subset A of a s...
 24.2.24.1.40: Using a Venn diagram, show that A <;;; B if and only if AU B = B.
Solutions for Chapter 24.2: Experiments, Outcomes, Events
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 24.2: Experiments, Outcomes, Events
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Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.)

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

Iterative method.
A sequence of steps intended to approach the desired solution.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Outer product uv T
= column times row = rank one matrix.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.