- 12-8.1: What is the probability that a randomly selected student spends mor...
- 12-8.2: What is the probability that a randomly selected student spends les...
- 12-8.3: If Marys cat has 4 kittens, what is the probability that at least 3...
- 12-8.4: What is the expected number of males in a litter of 6?
- 12-8.5: What is the probability that a randomly selected set of 4 tires wil...
- 12-8.6: What is the probability that a randomly selected set of tires will ...
- 12-8.7: What is the probability that at least 20 of the flowers will be blue?
- 12-8.8: What is the expected number of white irises in Dans garden?
- 12-8.9: x > 1.5
- 12-8.10: x > 3
- 12-8.11: x > _1 4
- 12-8.12: x < 1
- 12-8.13: x < _1 3
- 12-8.14: x < 2.5
- 12-8.15: What is the probability that there will be at least 12 successes?
- 12-8.16: What is the probability that there will be 12 failures?
- 12-8.17: What is the expected number of successes?
- 12-8.18: What is the probability that a randomly chosen bulb will last more ...
- 12-8.19: What is the probability that a randomly chosen bulb will last less ...
- 12-8.20: There is an 80% chance that a randomly chosen light bulb will last ...
- 12-8.21: What is the probability that more than 3 jurors will be men?
- 12-8.22: What is the probability that fewer than 6 jurors will vote to convict?
- 12-8.23: What is the expected number of votes for conviction?
- 12-8.24: OPEN ENDED Sketch the graph of an exponential distribution function...
- 12-8.25: REASONING An exponential distribution function has a mean of 2. A f...
- 12-8.26: CHALLENGE The average amount of money spent per day by students in ...
- 12-8.27: Writing in Math Your school has received a grant, and the administr...
- 12-8.28: ACT/SAT In rectangle ABCD, what is x + y in terms of z? A 90 + z B ...
- 12-8.29: REVIEW Your gym teacher is randomly distributing 15 red dodge balls...
- 12-8.30: What percent of the data lies between 39 and 61?
- 12-8.31: What is the probability that a data value selected at random is gre...
- 12-8.32: P(even)
- 12-8.33: P (1 or 6)
- 12-8.34: P(prime number)
- 12-8.35: (x - 7)(x + 9)
- 12-8.36: (4b2 + 7)2
- 12-8.37: (3q - 6)-(q + 13) + (-2q + 11)
- 12-8.38: third term of (a + b)7
- 12-8.39: fourth term of (c + d)8
- 12-8.40: fifth term of (x + y)9
Solutions for Chapter 12-8: Exponential and Binomial Distribu
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
peA) = det(A - AI) has peA) = zero matrix.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Remove row i and column j; multiply the determinant by (-I)i + j •
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
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.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
A sequence of steps intended to approach the desired solution.
= Xl (column 1) + ... + xn(column n) = combination of columns.
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.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Solvable system Ax = b.
The right side b is in the column space of A.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.