- 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: For Exercises 914, use the following information. An exponential di...
- 12.8.10: For Exercises 914, use the following information. An exponential di...
- 12.8.11: For Exercises 914, use the following information. An exponential di...
- 12.8.12: For Exercises 914, use the following information. An exponential di...
- 12.8.13: For Exercises 914, use the following information. An exponential di...
- 12.8.14: For Exercises 914, use the following information. An exponential di...
- 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: Sketch the graph of an exponential distribution function. Describe ...
- 12.8.25: An exponential distribution function has a mean of 2. A fellow stud...
- 12.8.26: The average amount of money spent per day by students in Mrs. Rosss...
- 12.8.27: Your school has received a grant, and the administration is conside...
- 12.8.28: In rectangle ABCD, what is x + y in terms of z?
- 12.8.29: Your gym teacher is randomly distributing 15 red dodge balls and 10...
- 12.8.30: A set of 260 data values is normally distributed with a mean of 50 ...
- 12.8.31: A set of 260 data values is normally distributed with a mean of 50 ...
- 12.8.32: A die is rolled, Find each probability.
- 12.8.33: A die is rolled, Find each probability.
- 12.8.34: A die is rolled, Find each probability.
- 12.8.35: Simplify each expression
- 12.8.36: Simplify each expression
- 12.8.37: Simplify each expression
- 12.8.38: Find the indicated term of each expression
- 12.8.39: Find the indicated term of each expression
- 12.8.40: Find the indicated term of each expression
Solutions for Chapter 12.8: Exponential and Binomial Distribution
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving | 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.
Remove row i and column j; multiply the determinant by (-I)i + j •
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.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
A sequence of steps intended to approach the desired solution.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
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.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
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 .
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).
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).
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Outer product uv T
= column times row = rank one matrix.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).