- 220.127.116.11.1: Suppose that E and F are events in a sample space and p(E) = 1/3, p...
- 18.104.22.168.2: Suppose that E and F are events in a sample space and p(E) = 2/3, p...
- 22.214.171.124.3: Suppose that Frida selects a ball by first picking one of two boxes...
- 126.96.36.199.4: Suppose that Ann selects a ball by first picking one of two boxes a...
- 188.8.131.52.5: Suppose that 8% of all bicycle racers use steroids, that a bicyclis...
- 184.108.40.206.6: When a test for steroids is given to soccer players, 98% of the pla...
- 220.127.116.11.7: Suppose that a test for opium use has a 2% false positive rate and ...
- 18.104.22.168.8: Suppose that one person in 10,000 people has a rare genetic disease...
- 22.214.171.124.9: Suppose that 8% of the patients tested in a clinic are infected wit...
- 126.96.36.199.10: Suppose that 4% of the patients tested in a clinic are infected wit...
- 188.8.131.52.11: An electronics company is planning to introduce a new camera phone....
- 184.108.40.206.12: A space probe near Neptune communicates with Earth using bit string...
- 220.127.116.11.13: A space probe near Neptune communicates with Earth using bit string...
- 18.104.22.168.14: Suppose that E, FJ , F2 , and F3 are events from a sample space S a...
- 22.214.171.124.15: In this exercise we will use Bayes' Theorem to solve the Monty Hall...
- 126.96.36.199.16: Ramesh can get to work three different ways: by bicycle, by car, or...
- 188.8.131.52.17: Prove Theorem 2, the extended form of Bayes' Theorem. That is, supp...
- 184.108.40.206.18: Suppose that a Bayesian spam filter is trained on a set of 500 spam...
- 220.127.116.11.19: Suppose that a Bayesian spam filter is trained on a set of 1000 spa...
- 18.104.22.168.20: Would we reject a message as spam in Example 4 a) using just the fa...
- 22.214.171.124.21: Suppose that a Bayesian spam filter is trained on a set of 10,000 s...
- 126.96.36.199.22: Suppose that we have prior information concerning whether a random ...
- 188.8.131.52.23: Suppose that EI and E2 are the events that an incoming mail message...
Solutions for Chapter 6.3: Discrete Probability
Full solutions for Discrete Mathematics and Its Applications | 6th Edition
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.
Column space C (A) =
space of all combinations of the columns of A.
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.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
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.
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].
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.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
A sequence of steps intended to approach the desired solution.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
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.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.