Discrete Mathematics With Applications
5th Edition
ISBN: 9781337694193
Author: EPP, Susanna S.
Publisher: Cengage Learning,
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Chapter 6.4, Problem 26ES
Some English adjectives are descriptive of themselves (for instance, the word polysyllabic is polysyllabic) whereas other are not monosyllabic). The word monosyllabic is not monosyllabic). The word monosyllabici refers to an monosyllabic). The word herterological refers to an adjective that does not describe itslf. Is heterological hererological? Exaplain your answer.
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Module Code: MATH380202
3. (a) Let {} be a white noise process with variance σ2.
Define an ARMA(p,q) process {X} in terms of {+} and state (without proof)
conditions for {X} to be (i) weakly stationary and (ii) invertible.
Define what is meant by an ARIMA (p, d, q) process. Let {Y} be such an ARIMA(p, d, q)
process and show how it can also be represented as an ARMA process, giving the
AR and MA orders of this representation.
(b) The following tables show the first nine sample autocorrelations and partial auto-
correlations of X and Y₁ = VX+ for a series of n = 1095 observations. (Notice
that the notation in this part has no relationship with the notation in part (a) of
this question.)
Identify a model for this time series and obtain preliminary estimates for the pa-
rameters of your model.
X₁
= 15.51, s² = 317.43.
k
1
2
3
4
5
6
7
Pk
0.981
0.974
0.968
akk 0.981 0.327
8
9
0.927
0.963 0.957 0.951 0.943 0.935
0.121 0.104 0.000 0.014 -0.067 -0.068 -0.012
Y₁ = VX : y = 0.03, s² = 11.48.
k
1…
Let G be a graph with n ≥ 2 vertices x1, x2, . . . , xn, and let A be the adjacency matrixof G. Prove that if G is connected, then every entry in the matrix A^n−1 + A^nis positive.
Module Code: MATH380202
1. (a) Define the terms "strongly stationary" and "weakly stationary".
Let {X} be a stochastic process defined for all t € Z. Assuming that {X+} is
weakly stationary, define the autocorrelation function (acf) Pk, for lag k.
What conditions must a process {X+) satisfy for it to be white noise?
(b) Let N(0, 1) for t€ Z, with the {+} being mutually independent. Which of
the following processes {X+} are weakly stationary for t> 0? Briefly justify your
answers.
i. Xt for all > 0.
ii. Xo~N(0,) and X₁ = 2X+-1+ &t for t > 0.
(c) Provide an expression for estimating the autocovariance function for a sample
X1,..., X believed to be from a weakly stationary process. How is the autocor-
relation function Pk then estimated, and a correlogram (or acf plot) constructed?
(d) Consider the weakly stationary stochastic process ✗+ = + + +-1+ +-2 where
{E} is a white noise process with variance 1. Compute the population autocorre-
lation function Pk for all k = 0, 1, ....
Chapter 6 Solutions
Discrete Mathematics With Applications
Ch. 6.1 - The notation is read”______” and means that___Ch. 6.1 - To use an element argument for proving that a set...Ch. 6.1 - Prob. 3TYCh. 6.1 - An element x is in AB if , and only if,_______Ch. 6.1 - An element x in AB if, and only if,______Ch. 6.1 - An element x is in B-A if, and only if,______Ch. 6.1 - An elements x is in Acif, and only if.______Ch. 6.1 - The empty set is a set with ______Ch. 6.1 - The power set of a set A is _____Ch. 6.1 - Prob. 10TY
Ch. 6.1 - A collection of nonempty set is a partition of a...Ch. 6.1 - Prob. 1ESCh. 6.1 - Complete the proof from Example 6.1.3: Prove that...Ch. 6.1 - Let sets R, S, and T be defined as follows:...Ch. 6.1 - Let A={nZn=5rforsomeintegerr} and...Ch. 6.1 - Prob. 5ESCh. 6.1 - Let...Ch. 6.1 - ...Ch. 6.1 - Prob. 8ESCh. 6.1 - Complete the following sentences without using the...Ch. 6.1 - ...Ch. 6.1 - Let the universal set be R, the set of all real...Ch. 6.1 - Let the universal set be R, the set of all real...Ch. 6.1 - Let S be the set of all strings of 0’s and 1’s of...Ch. 6.1 - Prob. 14ESCh. 6.1 - Prob. 15ESCh. 6.1 - Prob. 16ESCh. 6.1 - Prob. 17ESCh. 6.1 - a. Is the number 0 in ? Why? b. Is ={} ? Why ? c....Ch. 6.1 - Prob. 19ESCh. 6.1 - Let Bi={xR0xi} for each integer i=1,2,3,4. a....Ch. 6.1 - Let Ci={i,i} for each nonnegative integer i.Ch. 6.1 - Let Di={xR-ixi}=[i,i] for each nonnegative integer...Ch. 6.1 - Let Vi={xR1ix1i}=[1i,1i] for each positive integer...Ch. 6.1 - Let Wi={xRxi}=(i,) for each nonnegative integer i....Ch. 6.1 - Let Ri={xR1x1+1i}=[1,1+1i]foreachpositiveintegeri....Ch. 6.1 - Let Si={xR1x1+1i}=(1,1+1i) for each positive...Ch. 6.1 - Prob. 27ESCh. 6.1 - Let E be the set of all even integers and O the...Ch. 6.1 - Let R be the set of all real number. Is a...Ch. 6.1 - Let Z be the set of all integers and let...Ch. 6.1 - Prob. 31ESCh. 6.1 - Suppose A={1} and B={u,v} . Find P(AB) . Suppose...Ch. 6.1 - Find P() FindP(p()). Find p(p(p())) .Ch. 6.1 - Prob. 34ESCh. 6.1 - Prob. 35ESCh. 6.1 - Prob. 36ESCh. 6.1 - Prob. 37ESCh. 6.1 - Write an algorithm to determine whether a given...Ch. 6.2 - Prob. 1TYCh. 6.2 - Prob. 2TYCh. 6.2 - Prob. 3TYCh. 6.2 - Prob. 4TYCh. 6.2 - Prob. 5TYCh. 6.2 - Prob. 6TYCh. 6.2 - To say that an element is in A(BC) means that it...Ch. 6.2 - The following are two proofs that for all sets A...Ch. 6.2 - In 3 and 4, supply explanations of the steps in...Ch. 6.2 - Prob. 4ESCh. 6.2 - Prob. 5ESCh. 6.2 - Let and stand for the words “intersection” and...Ch. 6.2 - Prob. 7ESCh. 6.2 - Prob. 8ESCh. 6.2 - Prob. 9ESCh. 6.2 - Prob. 10ESCh. 6.2 - Prob. 11ESCh. 6.2 - Prob. 12ESCh. 6.2 - Prob. 13ESCh. 6.2 - Prob. 14ESCh. 6.2 - Prob. 15ESCh. 6.2 - Prob. 16ESCh. 6.2 - Prob. 17ESCh. 6.2 - Prob. 18ESCh. 6.2 - Prob. 19ESCh. 6.2 - Prob. 20ESCh. 6.2 - Prob. 21ESCh. 6.2 - Prob. 22ESCh. 6.2 - Prob. 23ESCh. 6.2 - Prob. 24ESCh. 6.2 - Prob. 25ESCh. 6.2 - Prob. 26ESCh. 6.2 - Fill in the blanks in the following proof that for...Ch. 6.2 - Prob. 28ESCh. 6.2 - Prob. 29ESCh. 6.2 - Prob. 30ESCh. 6.2 - Prob. 31ESCh. 6.2 - Prob. 32ESCh. 6.2 - Prob. 33ESCh. 6.2 - Prob. 34ESCh. 6.2 - Prob. 35ESCh. 6.2 - Prob. 36ESCh. 6.2 - Prob. 37ESCh. 6.2 - Prob. 38ESCh. 6.2 - Prove each statement is 39-44. For all sets A and...Ch. 6.2 - Prove each statement in 39-44. For every positive...Ch. 6.2 - Prob. 41ESCh. 6.2 - Prove each statement in 39-44. For every positive...Ch. 6.2 - Prob. 43ESCh. 6.2 - Prob. 44ESCh. 6.3 - Given a proposed set identity set identity...Ch. 6.3 - When using algebraic method for proving a set...Ch. 6.3 - Prob. 3TYCh. 6.3 - Prob. 1ESCh. 6.3 - Prob. 2ESCh. 6.3 - Prob. 3ESCh. 6.3 - Prob. 4ESCh. 6.3 - Prob. 5ESCh. 6.3 - Prob. 6ESCh. 6.3 - Prob. 7ESCh. 6.3 - Prob. 8ESCh. 6.3 - Prob. 9ESCh. 6.3 - Prob. 10ESCh. 6.3 - Prob. 11ESCh. 6.3 - Prob. 12ESCh. 6.3 - Prob. 13ESCh. 6.3 - Prob. 14ESCh. 6.3 - Prob. 15ESCh. 6.3 - Prob. 16ESCh. 6.3 - Prob. 17ESCh. 6.3 - Prob. 18ESCh. 6.3 - Prob. 19ESCh. 6.3 - Prob. 20ESCh. 6.3 - Prob. 21ESCh. 6.3 - Write a negation for each of the following...Ch. 6.3 - Let S={a,b,c} and for each integer i = 0, 1, 2, 3,...Ch. 6.3 - Let A={t,u,v,w} , and let S1 be the set of all...Ch. 6.3 - Prob. 25ESCh. 6.3 - Prob. 26ESCh. 6.3 - Prob. 27ESCh. 6.3 - Prob. 28ESCh. 6.3 - Some steps are missing from the following proof...Ch. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - Prob. 31ESCh. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - Prob. 33ESCh. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - In 30—40, construct an algebraic proof for the...Ch. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - In 30-40, construct an algebraic proof for the...Ch. 6.3 - Prob. 41ESCh. 6.3 - Prob. 42ESCh. 6.3 - Prob. 43ESCh. 6.3 - Prob. 44ESCh. 6.3 - Consider the following set property: For all sets...Ch. 6.3 - Prob. 46ESCh. 6.3 - Prob. 47ESCh. 6.3 - Prob. 48ESCh. 6.3 - Prob. 49ESCh. 6.3 - Prob. 50ESCh. 6.3 - Prob. 51ESCh. 6.3 - Prob. 52ESCh. 6.3 - Prob. 53ESCh. 6.3 - Prob. 54ESCh. 6.4 - In the comparison between the structure of the set...Ch. 6.4 - Prob. 2TYCh. 6.4 - Prob. 3TYCh. 6.4 - Prob. 1ESCh. 6.4 - Prob. 2ESCh. 6.4 - In 1-3 assume that B is a Boolean algebra with...Ch. 6.4 - Prob. 4ESCh. 6.4 - Prob. 5ESCh. 6.4 - Prob. 6ESCh. 6.4 - Prob. 7ESCh. 6.4 - Prob. 8ESCh. 6.4 - Prob. 9ESCh. 6.4 - In 4—10 assume that B is a Boolean algebra with...Ch. 6.4 - Prob. 11ESCh. 6.4 - Exercises 12-15 provide an outline for a proof...Ch. 6.4 - Prob. 13ESCh. 6.4 - Exercises 12-15 provide an outline for a proof...Ch. 6.4 - Prob. 15ESCh. 6.4 - Prob. 16ESCh. 6.4 - Prob. 17ESCh. 6.4 - In 16-21 determine where each sentence is a...Ch. 6.4 - In 16-21 determin whether each sentence is a...Ch. 6.4 - In 16-21 determine wherether each sentence is a...Ch. 6.4 - In 16-21 determine wherether each sentence is a...Ch. 6.4 - Prob. 22ESCh. 6.4 - Prob. 23ESCh. 6.4 - Can there exist a cimputer program that has as...Ch. 6.4 - Can there exist a book that refers to all those...Ch. 6.4 - Some English adjectives are descriptive of...Ch. 6.4 - As strange as it may seem, it is possible to give...Ch. 6.4 - Is there an alogroithm whichm for a fixed quantity...Ch. 6.4 - Prob. 29ES
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