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APPENDIX A
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Mathematical Topics
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Special Sums Integrals Euler s Formulas The Gamma Function The Beta Function
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APPENDIX B APPENDIX C APPENDIX D
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Ordinates y of the Standard Normal Curve at z Areas under the Standard Normal Curve from 0 to z Percentile Values tp for Student s t Distribution with n Degrees of Freedom Percentile Values x2 for the Chi-Square Distribution p with n Degrees of Freedom 95th and 99th Percentile Values for the F Distribution with n1, n2 Degrees of Freedom Values of e2l Random Numbers
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413 414
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417 419 419 420 423
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APPENDIX G APPENDIX H
SUBJECT INDEX INDEX FOR SOLVED PROBLEMS
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PART I
Probability
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CHAPTER 12 CHAPTER 1
Basic Probability
Random Experiments
We are all familiar with the importance of experiments in science and engineering. Experimentation is useful to us because we can assume that if we perform certain experiments under very nearly identical conditions, we will arrive at results that are essentially the same. In these circumstances, we are able to control the value of the variables that affect the outcome of the experiment. However, in some experiments, we are not able to ascertain or control the value of certain variables so that the results will vary from one performance of the experiment to the next even though most of the conditions are the same. These experiments are described as random. The following are some examples.
EXAMPLE 1.1 If we toss a coin, the result of the experiment is that it will either come up tails, symbolized by T (or 0), or heads, symbolized by H (or 1), i.e., one of the elements of the set {H, T} (or {0, 1}). EXAMPLE 1.2 If we toss a die, the result of the experiment is that it will come up with one of the numbers in the set {1, 2, 3, 4, 5, 6}. EXAMPLE 1.3 If we toss a coin twice, there are four results possible, as indicated by {HH, HT, TH, TT}, i.e., both heads, heads on first and tails on second, etc. EXAMPLE 1.4 If we are making bolts with a machine, the result of the experiment is that some may be defective. Thus when a bolt is made, it will be a member of the set {defective, nondefective}. EXAMPLE 1.5 If an experiment consists of measuring lifetimes of electric light bulbs produced by a company, then the result of the experiment is a time t in hours that lies in some interval say, 0 t 4000 where we assume that no bulb lasts more than 4000 hours.
Sample Spaces
A set S that consists of all possible outcomes of a random experiment is called a sample space, and each outcome is called a sample point. Often there will be more than one sample space that can describe outcomes of an experiment, but there is usually only one that will provide the most information.
EXAMPLE 1.6 If we toss a die, one sample space, or set of all possible outcomes, is given by {1, 2, 3, 4, 5, 6} while another is {odd, even}. It is clear, however, that the latter would not be adequate to determine, for example, whether an outcome is divisible by 3.
It is often useful to portray a sample space graphically. In such cases it is desirable to use numbers in place of letters whenever possible.
EXAMPLE 1.7 If we toss a coin twice and use 0 to represent tails and 1 to represent heads, the sample space (see Example 1.3) can be portrayed by points as in Fig. 1-1 where, for example, (0, 1) represents tails on first toss and heads on second toss, i.e., TH.
CHAPTER 1 Basic Probability
Fig. 1-1
If a sample space has a finite number of points, as in Example 1.7, it is called a finite sample space. If it has as many points as there are natural numbers 1, 2, 3, . . . , it is called a countably infinite sample space. If it has as many points as there are in some interval on the x axis, such as 0 x 1, it is called a noncountably infinite sample space. A sample space that is finite or countably infinite is often called a discrete sample space, while one that is noncountably infinite is called a nondiscrete sample space.
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