沙伦·罗斯《概率初步》第五版详解

1星需积分: 10 46 浏览量更新于2024-07-25 收藏 19.09MB PDF 举报

《沙伦·罗斯 - 首门概率论入门》第五版是Sheldon Ross所著的一本经典教材，由University of California, Berkeley的教授撰写，专为学习概率论的学生和专业人士设计。本书是Sheldon Ross系列教程中的一部，以其深入浅出的讲解和丰富的实例，在统计学和数学教育领域享有盛誉。该第五版教材包含了全面且系统的概率论基础理论，覆盖了概率的基本概念、随机变量、概率分布、独立事件、条件概率、期望与方差、极限定理、多元随机变量以及更高级的主题如大数定律、中心极限定理等。书中不仅注重理论解析，还强调实践应用，通过大量的习题和案例分析，帮助读者逐步掌握概率论的核心技巧和实际问题的解决方法。在内容编排上，Sheldon Ross以清晰的逻辑结构展开讨论，确保了从基础概率概念到复杂模型的渐进式学习过程。每章末尾通常会有复习题和习题，供读者巩固所学，并提供了解决实际问题所需的练习。此外，书中还附有详细的参考文献和索引，便于读者进一步探索相关领域的研究和发展。对于图书馆馆藏而言，这本书的《Library of Congress Cataloging-in-Publication Data》提供了必要的出版信息，包括作者Sheldon M. Ross、出版社Prentice Hall以及出版年份和ISBN号，方便图书馆进行分类和检索。此外，编辑团队的协作也体现了教材出版的专业性和严谨性，从 acquisitions editor 到 marketing manager 的多个角色共同确保了教材的质量和市场推广。《Sheldon Ross - A First Course in Probability》第五版是一本全面且实用的概率论入门指南，适合大学生、研究生、研究人员和任何希望系统学习概率论的人士使用，是提高概率理论素养的重要参考资料。无论是作为课堂教学材料还是自我学习的辅助读物，它都能提供扎实的基础知识和深入的理解。

Chapter

Combinatorial Analysis

when 0

12, and let it equal 0 otherwise. This quantity represents the number

of different subgroups of size i that can be chosen from a set of size

72. It is often

called a

birzo1ninl coejjficieizt because of its prominence in the binomid theorem,

which states that

.....

For nonnegative integers 121,

. .

12,

summing to n,

is the number of ways of dividing up

items into

distinct nonoverlapping

subgroups of sizes

121, n2,

. . .

n,.

PROBLEMS

(a) How many different 7-place license plates are possible if the first 2 places

are for letters and the other 5 for numbers?

(b)

Repeat part (a) under the assumption that no letter or number can be

repeated

a single license plate.

How many outcome sequences are possible when a die is rolled four times,

where we say, for instance, that the outcome is 3, 4, 3, 1 if the first roll

landed on 3, the second on 4, the third on 3, and the fourth on

Twenty workers are to be assigned to 20 different jobs, one to each job. How

many different assignments are possible?

John, Jim, Jay, and Jack have formed a band consisting of 4 instruments.

each of the boys can play

all

4 instruments, how many different arrangements

are possible? What if John and Jim can play all

instruments, but Jay and

Jack can each play only piano and drums?

For years, telephone area codes in the United States and Canada consisted

of a sequence of three digits. The first digit was an integer between 2 and 9;

the second digit was either

0 or 1; the third digit was any integer between 1

and 9. How many area codes were possible? How many area codes starting

with a 4 were possible?

well-known nursery rhyme starts as follows:

As I was going to

St.

Ives

met a

man

with 7 wives.

Each wife had-7 sacks.

Each sack had 7 cats.

Each cat had 7 kittens.

How many kittens did the traveler meet?

Problems

(a)

In how many ways can 3 boys,and 3 ,girls sit in a row?

(b)

In how many ways can 3 boys and 3 girls sit in a row if the boys and

the girls are each to sit together?

(d) In how many ways if no two people of the same sex are allowed to

sit together?

How many different letter arrangements can be made from the letters

""E;

(d) ARRANGE?

child has 12 blocks, of which-6 are black,

are red,

is white, and 1 is

blue.

the child puts the blocks in a line, how many arrangements are possible?

10.

In how many ways can

people be seated in a row if

(a) there are no restrictions on the seating arrangement;

(b)

persons

and

must sit next to each other;

each other;

(d) there are 5 men and they

must sit next to each other;

(e)

there are 4 married couples and each couple must sit together?

11.

In how many ways can 3 novels, 2 mathematics books, and 1 chemistry book

be arranged on a bookshelf

(a) the books can be arranged in any order;

(b)

the mathematics books must be together and the novels must be together;

any order?

12.

Five separate awards (best scholarship, best leadership qualities, and so on)

are to be presented to selected students from a class of 30. How many different

outcomes are possible

(a) a student can receive any number of awards;

(b)

each student can receive at most 1 award?

13.

Consider a group of 20 people.

everyone shakes hands with everyone else,

how many handshakes take place?

14.

How many 5-card poker hands are there?

15.

dance class consists of 22 students, 10 women and 12 men.

5 men and

5 women are to be chosen and then paired off, how many results are possible?

16.

student has to sell 2 books from a collection of

math. 7 science, and

economics books. How many choices are possible if

(a) both books are to be on the same subject;

(b)

the books are to be on different subjects?

17.

total of 7 different gifts are to be distributed among 10 children. How

many distinct results are possible

no child is to receive more than one gift?

18.

committee of 7, consisting of 2 Republicans, 2 Democrats, qnd 3 Indepen-

dents, is to be chosen from a group of

Republicqs,

Democrats, and 4

Independents. How many committees are possible?

Chapter

Cornbinatonal Analysis

19.

From a group of

women and

men a committee consisting of 3 men and

3 women is to be formed. How many different committees are possible if

(a)

2 of the men refuse to serve together;

(b)

of the women refuse to serve together;

(c)

1 man and 1 woman refusp to serve together?

20.

person has

friends, of whom 5 will be invited to a party.

(a) How many choices are there if 2 of the friends are feuding and will not

attend together?

(b)

How many choices if 2 of the friends will only attend together?

21.

Consider the ,orid of points shown below. Suppose that starting at the point

labeled

you can go one step up or one step to the right at each move. This

is continued until the point labeled

is reached. How many different paths

from

are possible?

HINT:

Note that to reach

from

you must take

steps to the right and

3 steps upward.

22.

Problem 21, how many different paths are there from

that go through

the point circled below?

Theoretical Exercises

24.

Expand (3x'

y)5.

25.

The game of bridge is played by

players, each of whom is dealt 13 cards.

How many bridge deals are possible?

26.

Expand (xl

2x2

3x314.

27.

12 people are to be divided into 3 committees of respective sizes 3,

and

5, how many divisions are possible?

28.

new teachers are to be divided among

schools, how many divisions

are possible?

What if each school must receive 2 teachers?

29.

Ten weight lifters are competing in a team weight-lifting contest. Of the

lifters, 3 are from the United States,

are from Russia,

are from China,

and 1 is from Canada.

the scoring takes account of the countries that the

Lifters represent but not their individual identities, how many different out-

comes are possible from the point of view of scores? How many different

outcomes correspond to results in which the United States has 1 competitor

in the

to^

three and 2 in the bottom three?

30.

Delegates from 10 countries, including Russia, France, England, and the

United States, are to be seated in a row. How many different seating arrange-

ments are possible if the French and English delegates are to be seated next

to each other, and the Russian and U.S. delegates are not to be next to

each other?

"31.

identical blackboards are to be divided among

schools, how many

divisions are possible? How many, if each school must receive at least 1 black-

board?

"32.

elevator starts at the basement with

people (not including the elevator

operator) and discharges them

all

by the time it reaches the top floor,

number

how many ways could the operator have perceived the people

leaving the elevator if

all

people look alike to him? What if the

people

consisted of 5 men and 3 women and the operator could tell a man from

a woman?

"33.

We have 20 thousand dollars that must be invested among

possible opportu-

nities. Each investment must be

integral in units of

thousand dollars, and

there are minimal investments that need to be made if one is to invest in

these opportunities. The minimal investments are 2, 2, 3, and

thousand

dollars. How many different investment strategies are available

(a) an investment must be made in each opportunity;

(b)

investments must be made in at least 3 of the

opportunities?

THEORETICAL EXERCISES

Prove the generalized version of the basic counting principle.

23.

psychology laboratory conducting dream research contains 3 rooms, with

Two experiments are to be performed. The first can result in any one of

2 beds in each room.

sets of identical twins are to be assigned to these

possible outcomes.

the first experiment results in outcome number

then

beds so that each set of twins sleeps in different beds in the same room,

the second experiment can result in any of

rzi

possible outcomes,

how many assignments are possible?

nz.

What is the number of possible outcomes of the two experiments?

-,.

Chapter

Combinatorid Analysis

how many ways can

objects be selected from a set of

if the order of

selection is considered relevant?

There are

(;)

different linear arrangements of

balls of which

are black

, ,

and

are white. Give a combinatorial explanation of this fact.

Determine the number of vectors (xl,

. . .

x,,), such that each xi is either

or 1 and

How many vectors xl,

. .

xk are there for which each

is a positive integer

such that 1

and

+ +

xk?

Give an analytic proof of Equation (4.1).

18.

Prove that.

Consider a group of n men and

women. How many groups of size

are possible?

Use

heo ore tical

Exercise

to prove that

10.

From a group of n people, suppose that we want to choose a committee of

k, k

n, one of whom is to be designated as chairperson.

(a) By focusing first on the choice of the committee and then on the choice

of the chair, argue that there are

k possible choices.

(b)

By focusing first on the choice of the nonchair committee members

and then on the choice of the chair, argue that there are

(11

possible choices.

(c)

By focusing first on the choice of the chair and then on the choice of

the other committee members, argue that there are

possible

choices.

(d)

Conclude from parts (a), (b), and (c) that

(e)

Use the factorial definition of

to verify the identity in part (d).

Theoretical Exercises

11.

The following identity is known as Fermat's combinatorial identity.

Give a combinatorial argument (no computations are needed) to establish this

identity.

HINT:

Consider the set of numbers 1 through n. How many subsets of size

k have

as their highest-numbered member?

12.

Consider the following combinatorial identity:

(a) Present a combinatorial argument for the above by considering a set of

n people and determining, in two ways, the number of possible selections

of a committee of any size and a chairperson for the committee.

HINT: (i) How many possible selections are there of a committee of

size k and its chairperson?

(ii) How many possible selections are there of a chairperson and

the other committee members?

(b)

Verify the following identity for n

1, 2,

For a combinatorial proof of the above, consider a set of

people, and

argue that both sides of the identity above represent the number of different

selections of a committee, its chairperson, and its secretary (possibly the

same as the chairperson).

HINT:

(i) How many different selections result

the committee con-

taining exactly k people?

(ii) How many different selections are there in which the chair-

person and the secretary are the same?

(ANSWER:

n2n- I.)

(iii) How many different selections result in the chairperson and

the secretary being different?

(c)

Now argue that

13.

Show that for

HINT:

Use the binomial theorem.

Chapter

Cornbinatonal Analysis

14.

From a set of

people a committee of size

is to be chosen, and from this

committee a subcommittee of size

j, is also to be chosen.

(a) Derive a combinatorial identity by computing, in two ways, the number

of possible choices of the committee and subcommittee-first by suppos-

ing that the committee is chosen first and then the subcommittee, and

second by supposing that the subcommittee is chosen first and then the

remaining members of the committee are chosen.

(b) Use part (a) to prove the following combinatorial identity:

15.

Let Hk(rz) be the number of vectors xl,

xk for which each

is a positive

integer satisfying 1

and xl

. . .

xk.

(a) Without any computations, argue that

m~:

How many vectors are there in which xk

(b)

Use the preceding recursion to compute H3(5).

HINT:

First compute H2(rz) for

1, 2, 3,

16.

Consider a tournament of

contestants in which the outcome is an ordering

of these contestants, with ties allowed. That is, the outcome partitions the

players into groups, with the first group consisting of the players that tied

for first place, the next group being those that tied for the next best position,

and so on. Let

N(rz) denote the number of different possible outcomes. For

instance, N(2)

3 since in a tournament with 2 contestants, player 1 could

be uniquely first, player 2 could be uniquely first, or they could tie for first.

(a) List all the possible outcomes when

(b)

With N(0) defined to equal 1, argue, without any computations, that

HINT:

HOW many outcomes are there in which

players tie for last place?

(b)

is equivalent to the following:

(d) Use the recursion to find N(3) and

N(4).

Self-test Problems

and

Exercises

17.

Present a combinatorial explanation of why

)

18.

Argue that

HINT:

Use an argument similar to the one used to establish Equation (4.1).

19.

Prove the multinomial theorem.

"20.

how many ways can

identical balls be distributed into

I-

urns so that the

ith urn contains at least

r?zi

balls, for each

I-?

Assume that

"21.

Argue that there are exactly

(I])

solutions of

for

whi* exactly

of the xi are equal to

"22.

Consider a function f(xl,

. .

x,:,,) of

variables. How many different partial

derivatives of order

I-

does it possess?

"23.

Determine the number of vectors (xl,

. . .

x,,), such that eachxi is a nonnegative

integer and

SELF-TEST PROBLEMS

AND

EXERCISES

How many different linear arrangements are there of the letters A,

C, D,

E, F for which

(a)

A and

are next to each other;

(b) A is before

and

is before C;

(d) A is before

and C is before D;

(e)

A and

are next to each other and C and D are also next to each other;

(f)

is not last in line?

4 Americans, 3 Frenchmen, and

Englishmen are to be seated in a row,

how many seating arrangements are possible when people of the same national-

ity must sit next to each other?

Chapter

Combinatorial Analysis

A president, treasurer, and secretary, all different, are to be chosen from a

club consisting of 10 people. How many different choices of officers are

possible if

(a)

there are no restrictions;

(b)

and

will not serve together;

(c)

and

will serve together or not at all;

(d)

must be an officer;

(e)

will serve only

he is president?

A student is to answer

out of 10 questions in an examination. How many

choices has she? How many

she must answer at least

of the first

ques-

tions?

In how many ways can a man divide

gifts among his

children

the eldest

is to receive

gd3s and the others 2 each?

How many different 7-place license plates are possible when

of the entries

are letters and 4 are digits? Assume that repetition of letters and numbers is

allowed and that there is no restriction on where the letters or numbers can

be placed.

Give a combinatorial explanation of the identity

Consider n-digit numbers where each digit is one of the 10 integers 0,

. . .

How many such numbers are there for which

(a) no two consecutive digits are equal;

(b)

0 appears as a digit a total of

times,

. . .

lz?

Consider three classes, each consisting of

students. From this group of

312

students, a group of

students is to be chosen.

(a) How many choices are possible?

(b)

How many choices are there in which all

students are in the same class?

of the

students are in the same

class and the other student is in a different class?

(d) How many choices are there in which all

students are in different classes?

(e) Using the results of parts (a) through (d), write a combinatorial identity.

*lo.

art collection on auction consisted of 4 Dalis,

van Goghs, and 6 Picassos.

At the auction were

art collectors.

a reporter noted only the number of

Dalis, van Goghs, and Picassos acquired by each collector, how many different

results could have been recorded if all works were sold?

*11.

Determine the number of vectors

(xl,

. .

x,)

such that each

is a positive

integer and

where

12.

Probability

2.1

INTRODUCTION

this chapter we introduce the concept of the probability of an event and

then show how these probabilities can be computed in certain situations. As a

preliminary, however, we need the concept of the sample space and the events

of an experiment.

2.2

SAMPLE SPACE AND EVENTS

Consider an experiment whose outcome is not predictable with certainty in ad-

vance. However, although the outcome of the experiment will not be known in

advance, let us suppose that the set of all possible outcomes is known. This set

all.possible outcomes of an experiment is known as the

sarnple space

of the

experiment and is denoted by

Some examples follow.

the outcome of an experiment consists in the determination of the sex

of a

new&orn child, then

Is, b}.

where the outcome g means that the child is a girl and b that it is a boy.

the outcome of an experiment is the order of finish in a race among

the

horses having post positions 1,

6, 7, then

{all 7! permutations of (1, 2,

6, 7))

Woutcome (2,

1, 6,

4, 7) means, for instance, that the number 2 horse

comes in first, then the number

horse, then the number 1 horse, and so on.

the experiment consists of flipping two coins, then the sample space

consists of the following four points:

{(H,

H),

(H, TI, (T,

H),

(T,

T)}

剩余265页未读，继续阅读

goodpick

粉丝: 0
资源: 1

沙伦·罗斯《概率初步》第五版详解

a first course in probability ross 6th（解答）

Sheldon M. Ross - Introduction to Probability Models

A First Course In Probability 7Ed (Solution Manual Prentice Hall 147S) by Sheldon Ross

A first course in probability

A First Course In Probability 8th

概率论基础教程 Sheldon M．Ross 第八版 英文原版

【LSTM回归预测】基于emd结合长短记忆神经网络lstm实现风速回归预测附Matlab源码.rar

SVM classifier train test result

奇迹书屋_24.08.10.apk

Qt文件拷贝器（源码）

最新资源

概率论基础教程 Sheldon M．Ross 第八版英文原版