 125.1: For Exercises 13, use the table that shows the possible sums when r...
 125.2: For Exercises 13, use the table that shows the possible sums when r...
 125.3: For Exercises 13, use the table that shows the possible sums when r...
 125.4: GRADES For Exercises 46, use the table that shows a classs grade di...
 125.5: GRADES For Exercises 46, use the table that shows a classs grade di...
 125.6: GRADES For Exercises 46, use the table that shows a classs grade di...
 125.7: For Exercises 710, the spinner shown is spun three times. Write the...
 125.8: For Exercises 710, the spinner shown is spun three times. Find the ...
 125.9: For Exercises 710, the spinner shown is spun three times. Make a pr...
 125.10: For Exercises 710, the spinner shown is spun three times. Do all po...
 125.11: SALES For Exercises 1114, use the following information. A music st...
 125.12: SALES For Exercises 1114, use the following information. A music st...
 125.13: SALES For Exercises 1114, use the following information. A music st...
 125.14: SALES For Exercises 1114, use the following information. A music st...
 125.15: EDUCATION For Exercises 1517, use the table, which shows the educat...
 125.16: EDUCATION For Exercises 1517, use the table, which shows the educat...
 125.17: EDUCATION For Exercises 1517, use the table, which shows the educat...
 125.18: SPORTS For Exercises 18 and 19, use the graph that shows the sports...
 125.19: SPORTS For Exercises 18 and 19, use the graph that shows the sports...
 125.20: OPEN ENDED Describe reallife data that could be displayed in a pro...
 125.21: CHALLENGE Suppose a married couple keeps having children until they...
 125.22: Writing in Math Refer to the information on page 672 to explain how...
 125.23: The table shows the probability distribution for the number of head...
 125.24: REVIEW Mr. Perez works 40 hours a week at The Used Car Emporium. He...
 125.25: A card is drawn from a standard deck of 52 cards. Find each probabi...
 125.26: A card is drawn from a standard deck of 52 cards. Find each probabi...
 125.27: A card is drawn from a standard deck of 52 cards. Find each probabi...
 125.28: Evaluate. 10C7
 125.29: Evaluate. 12C5
 125.30: Evaluate. (6P3)(5P3)
 125.31: SAVINGS For Exercises 3132, use the following information. Selena i...
 125.32: SAVINGS For Exercises 3132, use the following information. Selena i...
 125.33: Which type of compounding appears more profitable? Explain.
 125.34: PREREQUISITE SKILL Write each fraction as a percent rounded to the ...
 125.35: PREREQUISITE SKILL Write each fraction as a percent rounded to the ...
 125.36: PREREQUISITE SKILL Write each fraction as a percent rounded to the ...
 125.37: PREREQUISITE SKILL Write each fraction as a percent rounded to the ...
 125.38: PREREQUISITE SKILL Write each fraction as a percent rounded to the ...
 125.39: PREREQUISITE SKILL Write each fraction as a percent rounded to the ...
Solutions for Chapter 125: Probability Distributions
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1)  1st Edition
ISBN: 9780078738227
Solutions for Chapter 125: Probability Distributions
Get Full SolutionsSince 39 problems in chapter 125: Probability Distributions have been answered, more than 35204 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1. Chapter 125: Probability Distributions includes 39 full stepbystep solutions. Algebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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.

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.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.