- Chapter 6.1: Let .S; P / be a sample space with S D f1; 2; 3; : : : ; 10g. For a...
- Chapter 6.2: Three dice are dropped at random into a frame where they sit snugly...
- Chapter 6.3: Let .S; P / be a sample space where S D f1; 2; 3; : : : ; 10g and P...
- Chapter 6.4: Ten children (five boys and five girls) are standing in line. Assum...
- Chapter 6.5: Thirteen cards are drawn (without replacement) from a standard deck...
- Chapter 6.6: In the card game blackjack, each card in the deck has a numerical v...
- Chapter 6.7: A standard deck of cards is shuffled. What is the probability that ...
- Chapter 6.8: Let A be an event for a sample space .S; P /. Under certain circums...
- Chapter 6.9: Two squares are chosen (with replacement) from among the 64 squares...
- Chapter 6.10: . Repeat the previous problem, this time assuming the squares are c...
- Chapter 6.11: An unfair coin is tossed twice in a row. What is the probability th...
- Chapter 6.12: Let A and B be events for a sample space .S; P /. Suppose that A B ...
- Chapter 6.13: Consider the sample space .S; P / where S D fa; b; cg and P .a/ D 0...
- Chapter 6.14: A card is drawn from a well-shuffled deck. Let X be the blackjack v...
- Chapter 6.15: Let X and Y be independent random variables defined on a common sam...
- Chapter 6.16: Simplified stock market. Suppose there are three kinds of days: GOO...
Solutions for Chapter Chapter 6: Probability
Full solutions for Mathematics: A Discrete Introduction | 3rd Edition
Tv = Av + Vo = linear transformation plus shift.
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.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
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 II· Condition numbers measure the sensitivity of the output to change in the input.
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.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Invert A by row operations on [A I] to reach [I A-I].
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
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.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
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.