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ISBN 10: 1118947088
ISBN 13:978-1118947081
Author: Kinney
“This is a well-written and impressively presented introduction to probability and statistics. The text throughout is highly readable, and the author makes liberal use of graphs and diagrams to clarify the theory.” – The Statistician
Thoroughly updated, Probability: An Introduction with Statistical Applications, Second Edition features a comprehensive exploration of statistical data analysis as an application of probability. The new edition provides an introduction to statistics with accessible coverage of reliability, acceptance sampling, confidence intervals, hypothesis testing, and simple linear regression. Encouraging readers to develop a deeper intuitive understanding of probability, the author presents illustrative geometrical presentations and arguments without the need for rigorous mathematical proofs.
The Second Edition features interesting and practical examples from a variety of engineering and scientific fields, as well as:
- Over 880 problems at varying degrees of difficulty allowing readers to take on more challenging problems as their skill levels increase
- Chapter-by-chapter projects that aid in the visualization of probability distributions
- New coverage of statistical quality control and quality production
- An appendix dedicated to the use of Mathematica® and a companion website containing the referenced data sets
Featuring a practical and real-world approach, this textbook is ideal for a first course in probability for students majoring in statistics, engineering, business, psychology, operations research, and mathematics. Probability: An Introduction with Statistical Applications, Second Edition is also an excellent reference for researchers and professionals in any discipline who need to make decisions based on data as well as readers interested in learning how to accomplish effective decision making from data.
Probability: An Introduction with Statistical Applications 2nd Table of contents:
Chapter 1 Sample Spaces and Probability
1.1 Discrete Sample Spaces
Definition
Example 1.1.1
Figure 1.1 Sample space for tossing two dice.
Example 1.1.2
Exercises 1.1
1.2 Events; Axioms of Probability
Definition
Axioms of Probability
Definition
Figure 1.2 Venn diagram showing the event A.
Figure 1.3 Venn diagram showing disjoint events A and B.
1.3 Probability Theorems
Theorem 1
Figure 1.4 Venn diagram showing arbitrary events A and B.
Theorem 2
Example 1.3.1
Theorem 3
Figure 1.5 Venn diagram showing events, A, B, and C.
Example 1.3.2
Theorem 4
Example 1.3.3
Exercises 1.3
1.4 Conditional Probability and Independence
Example 1.4.1
Example 1.4.2
Example 1.4.3
Theorem: (Law of Total Probability):
Example 1.4.4
Example 1.4.5
Theorem (Bayes’ theorem):
Figure 1.6 Diagram for Example 1.4.5.
Figure 1.7 A geometric view of Bayes’ theorem.
Example 1.4.6
Figure 1.8 AIDS example.
Figure 1.9 P(A|T+)as a function of P(A).
Figure 1.10 P(A|T+) as a function of the incidence rate, r, if p = 0.95.
Figure 1.11 P(A|T+) as a function of r, the incidence rate, and P(T + |A).
Example 1.4.7
Figure 1.12 Diagram for the Monty Hall problem.
Independence
Definition
Example 1.4.8
Exercises 1.4
1.5 Some Examples
Example 1.5.1 (The Birthday Problem)
Figure 1.13 The birthday problem as a function of n, the number of people in the group.
Example 1.5.2
Figure 1.14 Polynomial approximation to the birthday data.
Example 1.5.3 (Mowing the Lawn)
Figure 1.15 n marbles for the lawn mowing problem.
Exercises 1.5
1.6 Reliability of Systems
Series Systems
Figure 1.16 A series system of two components.
Parallel Systems
Example 1.6.1
Figure 1.17 A parallel system of two components.
Figure 1.18 Reliability of series and parallel systems.
Figure 1.19 System for Example 1.6.1.
Figure 1.20 Reliability surface for Example 1.6.1.
Figure 1.21 Contour plot for the surface in Figure 1.20.
Exercises 1.6
1.7 Counting Techniques
Figure 1.22 Tree diagram showing counting principle 1.
Example 1.7.1
Definition:
Example 1.7.2
Theorem 4
Proof
Example 1.7.3
Example 1.7.4
Example 1.7.5
Figure 1.23 P(Median = k) for a sample of size 3 chosen from 10 cars.
Example 1.7.6
Figure 1.24 P(Median = k) for a sample of 9 chosen from 100 cars.
Example 1.7.7 (Ken–Ken Puzzles)
Exercises 1.7
Chapter Review
Theorem (Law of Total Probability):
Theorem (Bayes’ Theorem):
Theorem
Problems for Review
Supplementary Exercises for Chapter 1
Chapter 2 Discrete Random Variables and Probability Distributions
Definition:
2.1 Random Variables
Example 2.1.1
Figure 2.1 Discrete uniform probability distribution.
Example 2.1.2
Example 2.1.3
Figure 2.2 Sums on two fair dice.
Figure 2.3 Sums on three fair dice.
Example 2.1.4
Example 2.1.5
Figure 2.4 Sums on two similarly loaded dice.
Figure 2.5 Sums on two differently loaded dice.
Example 2.1.6
Figure 2.6 Geometric distribution.
2.2 Distribution Functions
Example 2.2.1
Figure 2.7 Distribution function for one toss of a fair die.
Exercises 2.2
2.3 Expected Values of Discrete Random Variables
Expected Value of a Discrete Random Variable
Example 2.3.1
Example 2.3.2
Example 2.3.3
Example 2.3.4
Example 2.3.5
Variance of a Random Variable
Definition:
Figure 2.8 Two random variables with the same mean value.
Example 2.3.6
Example 2.3.7
Example 2.3.8
Tchebycheff’s Inequality
Theorem 1:
Proof
Exercises 2.3
2.4 Binomial Distribution
Example 2.4.1
Example 2.4.2
2.5 A Recursion
Figure 2.9 (a) Binomial distribution, n = 10, p = 1/2. (b) Binomial distribution, n = 100, p = 1/2.
Figure 2.10 Binomial distribution, n = 50, p = 3/4
The Mean and Variance of the Binomial
Example 2.5.1
Exercises 2.5
2.6 Some Statistical Considerations
Table 2.1 Exact probability of binomial intervals around the mean for various values of n and p
Figure 2.11 Some confidence intervals.
Exercises 2.6
2.7 Hypothesis Testing: Binomial Random Variables
Example 2.7.1
Figure 2.12 β as a function of p for Example 2.7.1.
Example 2.7
Exercises 2.2.7
2.8 Distribution of A Sample Proportion
Example 2.8.1
Exercises 2.8
2.9 Geometric and Negative Binomial Distributions
Figure 2.13 A negative binomial distribution.
Example 2.9.1 All Heads
Example 2.9.2
Example 2.9.3 Candy Jars
Figure 2.14 The candy jars problem for n = 15.
A Recursion
Figure 2.15 E(X) for the candy jars problem.
Figure 2.16 Variance in the candy jars problem.
Exercises 2.9
2.10 The Hypergeometric Random Variable: Acceptance Sampling
Acceptance Sampling
The Hypergeometric Random Variable
Figure 2.17 Hypergeometric distribution with N = 12, n = 3, and D = 4.
Figure 2.18 Hypergeometric distribution with N = 1000, n = 30, and D = 400.
Some Specific Hypergeometric Distributions
Exercises 2.10
2.11 Acceptance Sampling (Continued)
Example 2.11.1
Example 2.11.2
Figure 2.19 Effect of sample size, n, on a sampling plan.
Figure 2.20 Effect of quality in the lot on the probability of acceptance.
Producer’s and Consumer’s Risks
Figure 2.21 Some operating characteristic curves.
Average Outgoing Quality
Example 2.11.3
Example 2.11.4
Example 2.11.5
Figure 2.22 AOQ as a function of the number of unacceptable items in the lot.
Double Sampling
Figure 2.23 Probability lot is accepted in a double sampling plan.
Figure 2.24 Average outgoing quality for a double sampling plan.
Exercises 2.11
2.12 The Hypergeometric Random Variable: Further Examples
Example 2.12.1 A Lottery
Example 2.12.2 A Card Game
2.13 The Poisson Random Variable
Mean and Variance of the Poisson
Some Comparisons
Figure 2.25 (a) Poisson distribution with parameter 0.60. (b) Binomial distribution with n = 100, p = 0.03.
Figure 2.26 (a) Poisson distribution with parameter 3. (b) Binomial distribution with n = 100, p = 0.03.
Example 2.13.1
2.14 The Poisson Process
Example 2.14.1
Solution 1
Solution 2
Exercises 2.14
Chapter Review
Problems for Review
Supplementary Exercises for Chapter 2
Chapter 3 Continuous Random Variables and Probability Distributions
3.1 Introduction
Figure 3.1 The spinner.
Definition
Figure 3.2 Probability density function for the loaded wheel.
Figure 3.3 Probability density function for the rigged wheel.
Figure 3.4 Cumulative distribution function for the fair wheel.
Mean and Variance
Example 3.1.1
Example 3.1.2
A Word on Words
Exercises 3.1
3.2 Uniform Distribution
Figure 3.5 Uniform distribution on the interval [a, b].
Example 3.2.1
Example 3.2.2
Exercises 3.2
3.3 Exponential Distribution
Example 3.3.1
Mean and Variance
Figure 3.6 An exponential distribution.
Distribution Function
Example 3.3.2
Example 3.3.3
Example 3.3.4
3.4 Reliability
Hazard Rate
Figure 3.7 A “bathtub” hazard rate function.
Exercises 3.4
3.5 Normal Distribution
Figure 3.8 Standard normal probability density function.
Figure 3.9 Some normal probability density functions.
Figure 3.10 Revolving the standard normal curve around the y-axis.
Example 3.5.1
Example 3.5.2
Example 3.5.3
Exercises 3.5
3.6 Normal Approximation to the Binomial Distribution
Example 3.6.1
Figure 3.11 Binomial distribution, n = 500, p = 0.10.
Figure 3.12 A histogram for the binomial distribution, n = 500, p = 0.10.
Figure 3.13 Normal curve approximation for the binomial, n = 500, p = 0.10.
Exercises 3.6
3.7 Gamma and Chi-Squared Distributions
Figure 3.14 Waiting time for the second Poisson event.
Figure 3.15 A gamma distribution.
Figure 3.16 Some χ2 distributions.
Example 3.7.1
Figure 3.17 distribution.
Exercises 3.7
3.8 Weibull Distribution
Figure 3.18 Some Weibull distributions.
Figure 3.19 Some reliability functions.
Exercises 3.8
Chapter Review
Problems for Review
Supplementary Exercises For Chapter 3
Chapter 4 Functions of Random Variables; Generating Functions; Statistical Applications
4.1 Introduction
4.2 Some Examples of Functions of Random Variables
4.3 Probability Distributions of Functions of Random Variables
Example 4.3.1
Example 4.3.2
Example 4.3.3
Expectation of a Function of X
Example 4.3.4
Exercises 4.3
4.4 Sums of Random Variables I
Example 4.4.1
Example 4.4.2
Figure 4.1 Sum of two independent discrete uniform variables, n = 4.
Exercises 4.4
4.5 Generating Functions
Example 4.5.1
Figure 4.2 Probabilities for one fair die.
Figure 4.3 Probabilities for sums on two fair dice.
Figure 4.4 Probabilities for sums on 4 fair dice.
Figure 4.5 Probabilities for sums on 12 fair dice.
Exercises 4.5
4.6 Some Properties of Generating Functions
4.7 Probability Generating Functions for Some Specific Probability Distributions
Binomial Distribution
Poisson’s Trials
Figure 4.6 Probabilities for the total number of heads when a fair coin is tossed 20 times followed by 10 tosses of a loaded coin with p = 1/3.
Example 4.7.1
Geometric Distribution
Collecting Premiums in Cereal Boxes
Figure 4.7 Probabilities for the cereal box problem.
Exercises 4.7
4.8 Moment Generating Functions
Definition
Example 4.8.1
Example 4.8.2
Example 4.8.3
Exercises 4.8
4.9 Properties of Moment Generating Functions
Theorem:
Proof
Example 4.9.1
4.10 Sums of Random Variables—II
Example 4.10.1 Sums of Normal Random Variables
Example 4.10.2 Sums of Exponential Random Variables
Figure 4.8 Sum of two independent exponential random variables.
Figure 4.9 Sum of three independent exponential variables.
Exercises 4.10
4.11 The Central Limit Theorem
Theorem 1:
Example 4.11.1
Exercises 4.11
4.12 Weak Law of Large Numbers
Figure 4.10 Simulation illustrating the weak law of large numbers.
4.13 Sampling Distribution of the Sample Variance
Figure 4.11 Sampling distribution for sample variances.
Figure 4.12 A probability distribution suggested by Figure 4.11.
Figure 4.13 Distribution of sample variances chosen from a standard normal distribution.
Example 4.13.1
Figure 4.14 distribution.
Exercises 4.13
4.14 Hypothesis Tests and Confidence Intervals for a Single Mean
Example 4.14.1
Confidence Intervals, σ Known
Example 4.14.2
Example 4.14.3
Student’s t Distribution
Theorem:
Example 4.14.4
Figure 4.15 Student t distributions for 3, 8, and 20 degrees of freedom.
p Values
Figure 4.16 Some confidence intervals.
Example 4.14.5
Example 4.14.6
Exercises 4.14
4.15 Hypothesis Tests on Two Samples
Tests on Two Means
Example 4.15.1
Example 4.15.2
Example 4.15.3
Tests on Two Variances
Figure 4.17 Some F distributions.
Example 4.15.4
Exercises 4.15
4.16 Least Squares Linear Regression
Example 4.16.1
Example 4.16.2
Figure 4.18 Scatter plot of data.
Figure 4.19 Regression line and data points.
Exercises 4.16
4.17 Quality Control Chart for
Example 4.17.1
Figure 4.20 A control chart for sample means.
Table 4.1
Figure 4.21 Factors c4 for a quality control chart.
Exercises 4.17
Chapter Review
Problems for Review
Supplementary Exercises for Chapter 4
Chapter 5 Bivariate Probability Distributions
5.1 Introduction
5.2 Joint and Marginal Distributions
Example 5.2.1
Table 5.1 Joint distribution for the coin tossing example
Figure 5.1 Scatter plot for the coin tossing example.
Definition:
Figure 5.2 Marginal distribution for Y in Example 5.2.1.
Example 5.2.2
Figure 5.3 Surface for Example 5.2.2.
Figure 5.4 Marginal distribution for X, Example 5.2.2.
Figure 5.5 Marginal distribution for Y, Example 5.2.2.
Example 5.2.3
Exercises 5.2
5.3 Conditional Distributions and Densities
Definition:
Example 5.3.1
Example 5.3.2
Figure 5.6 Sample space for f(x, y) = 2e− x − y, x ≥ 0, y ≥ x.
Figure 5.7 Probability surface for Example 5.3.2.
Exercises 5.3
5.4 Expected Values and the Correlation Coefficient
Definition:
Definition:
Definition:
Example 5.4.1
5.5 Conditional Expectations
Example 5.5.1
Example 5.5.2
Exercises 5.5
5.6 Bivariate Normal Densities
Figure 5.8 Normal probability surface, ρ = 0.
Figure 5.9 Normal bivariate surface, ρ = 0.5.
Figure 5.10 Normal bivariate surface, ρ = 0.9.
Contour Plots
Figure 5.11 Circular contours for a normal probability surface, ρ = 0.
Figure 5.12 Elliptical contours for a normal probability surface, ρ = 0.9.
Exercises 5.6
5.7 Functions of Random Variables
Example 5.7.1
Figure 5.13 Jointly distributed uniform variables.
Example 5.7.2
Figure 5.14 Sample space for two independent exponential variables.
Example 5.7.3
Figure 5.15 Sample space for the product of two uniform random variables.
Figure 5.16 Sample space for the quotient of two uniform random variables.
Example 5.7.4
Exercises 5.7
Chapter Review
Problems for Review
Supplementary Exercises for Chapter 5
Chapter 6 Recursions and Markov Chains
6.1 Introduction
6.2 Some Recursions and their Solutions
Example 6.2.1
Figure 6.1 a10 for the quality control inspector.
Figure 6.2 an for p = 1/4.
Solution of the Recursion (6.3)
Example 6.2.2
Figure 6.3 bn for a fair coin.
Figure 6.4 bn for a coin with p = 3/4.
Mean and Variance
Example 6.2.3
Exercises 6.2
6.3 Random Walk and Ruin
Figure 6.5 Probability of winning the gamblers’ ruin with an initial fortune of $10 against an opponent with an initial fortune of $N with p = 0.49.
Figure 6.6 Probability of winning the gambler’s ruin. The player has an initial fortune of $g. p is the probability an individual game is won. The opponent initially has $30.
Figure 6.7 A contour plot of the gambler’s ruin problem.
Expected Duration of the Game
Figure 6.8 Expected duration of the gambler’s ruin when the gambler initially has $g and the house has $(100 − g).
Exercises 6.3
6.4 Waiting Times for Patterns in Bernoulli Trials
Example 6.4.1
Generating Functions
Average Waiting Times
Means and Variances by Generating Functions
Exercises 6.4
6.5 Markov Chains
Example 6.5.1
Example 6.5.2
Definition
Theorem:
Example 6.5.3
Example 6.5.4
Example 6.5.5
Example 6.5.6
Example 6.5.7
Example 6.5.8
Exercises 6.5
Chapter Review
Problems for Review
Supplementary Exercises For Chapter 6
Chapter 7 Some Challenging Problems
7.1 My Socks and
7.2 Expected Value
7.3 Variance
7.4 Other “Socks” Problems
7.5 Coupon Collection and Related Problems
Three Prizes
Permutations
An Alternative Approach
Altering the Probabilities
A General Result
Expectations and Variances
Geometric Distribution
Variances
Waiting for Each of the Integers
Conditional Expectations
Other Expected Values
Waiting for All the Sums on Two Dice
7.6 Conclusion
7.7 Jackknifed Regression and the Bootstrap
Jackknifed Regression
Figure 7.1 Data and least squares regression line.
Table 7.1
Table 7.2
Table 7.3
Table 7.4
7.8 Cook’s Distance
7.9 The Bootstrap
Example 7.9.1
Figure 7.2 A gamma distribution.
Figure 7.3 Medians of bootstrap samples.
Example 7.9.2
Figure 7.4 Range of bootstrap samples.
7.10 On Waldegrave’s Problem
Three Players
7.11 Probabilities of Winning
7.12 More than Three Players
r + 1 Players
Probabilities of Each Player
Expected Length of the Series
Fibonacci Series
7.13 Conclusion
7.14 On Huygen’s First Problem
7.15 Changing the Sums for the Players
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