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f 1x1 2, f 1x2 2, p , f 1xn 2, respectively, and by independence the joint probability density function of the random sample is fX1 X2 p Xn 1x1, x2, p , xn 2 f 1x1 2 f 1x2 2 p f 1xn 2. De nition The random variables X1, X2, p , Xn are a random sample of size n if (a) the Xi s are independent random variables, and (b) every Xi has the same probability distribution.

To illustrate this de nition, suppose that we are investigating the effective service life of an electronic component used in a cardiac pacemaker and that component life is normally distributed. Then we would expect each of the observations on component life X1, X2, p , Xn in a random sample of n components to be independent random variables with exactly the same normal distribution. After the data are collected, the numerical values of the observed lifetimes are denoted as x1, x2, p , xn. The primary purpose in taking a random sample is to obtain information about the unknown population parameters. Suppose, for example, that we wish to reach a conclusion about the proportion of people in the United States who prefer a particular brand of soft drink. Let p represent the unknown value of this proportion. It is impractical to question every individual in the population to determine the true value of p. In order to make an inference regarding the true proportion p, a more reasonable procedure would be to select a random sample (of an appropriate size) and use the observed proportion p of people in this sample favoring the brand of soft drink. The sample proportion, p is computed by dividing the number of individuals in the sam ple who prefer the brand of soft drink by the total sample size n. Thus, p is a function of the observed values in the random sample. Since many random samples are possible from a population, the value of p will vary from sample to sample. That is, p is a random variable. Such a random variable is called a statistic. De nition A statistic is any function of the observations in a random sample.

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Gain in IQ points per decade Nation Japan (status 34 ) Austria (status 4) West Germany (status 34 ) France (status 4)a United States (status 2)b Mean Performance 7.3 9.3 9.3 6.3 3.3 7.1 Verbal 6.7 6.7 4.0 1.0 2.7 4.2 Full Scale 8.4 8.2 7.4 3.7 3.0 6.1

Entity = Data entity Relationship = Data relationship e.g., Physical Data Model e.g., System design

We have encountered statistics before. For example, if X, X2, p , Xn is a random sample of size n, the sample mean X, the sample variance S 2, and the sample standard deviation S are statistics. Although numerical summary statistics are very useful, graphical displays of sample data are a very powerful and extremely useful way to visually examine the data. We now present a few of the techniques that are most relevant to engineering applications of probability and statistics.

aData for France do not match the data in Figure 2.1; these Wechsler data are status 4 data that were excluded from that figure. bData for the U.S. are just WISC vs. WISC-R in order to hold generation (post-1950) constant.

In some cases, the conditional probability of P1B A2 might equal P(B). In this special case, knowledge that the outcome of the experiment is in event A does not affect the probability that the outcome is in event B.

EXAMPLE 2-23

Gains in IQ per decade for numerous nations, based either on verbal or nonverbal tests Nation Gain on Nonverbal Tests 12.5 10.0 9.3 9.3 7.6 7.3 6.7 6.0 4.9 4.4 4.2 4.0 3.6 3.3 6.4 Median Verbal Gain 4.1 Nation Japan Austria (Vienna) Canada (Saskatchewan) Scotland Belgium West Germany France New Zealand Norway Gain on Verbal Tests 6.7 6.7 4.9 4.3 4.1 4.0 3.7 2.4 2.2

Process = Application function I/O = User views Node = I/S function (processor, storage, etc.) Link = Line characteristics

Suppose a day s production of 850 manufactured parts contains 50 parts that do not meet customer requirements. Suppose two parts are selected from the batch, but the rst part is replaced before the second part is selected. What is the probability that the second part is defective (denoted as B) given that the rst part is defective (denoted as A) The probability needed can be expressed as P1B A2. Because the rst part is replaced prior to selecting the second part, the batch still contains 850 parts, of which 50 are defective. Therefore, the probability of B does not depend on whether or not the rst part was defective. That is, P1B A2 50 850

East Germany (Leipzig) France Austria (Vienna) West Germany Belgium Japan Netherlands Israel Australia China Norway Canada (Edmonton) Great Britain United States Median Nonverbal Gain

Also, the probability that both parts are defective is P1A B2 P1B 0 A2P1A2 a 50 50 b a b 850 850 0.0035

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