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1 1 1 1 1 52 52 52 52 13 This could also have been achieved from the sample space of Problem 1.1(a) where each sample point, in particular ace, has probability 1 > 13. It could also have been arrived at by simply reasoning that there are 13 numbers and so each has probability 1 > 13 of being drawn. 1 (b) P(11 > H) 52 1 1 1 (c) P(3 > C or 6 > D) P(3 > C ) P(6 > D) 52 52 26 1 1 1 13 1 c (d) P(H) P(1 > H or 2 > H or c13 > H) 52 52 52 52 4 This could also have been arrived at by noting that there are four suits and each has equal probability1>2 of being drawn. 1 3 (e) P(Hr) 1 P(H) 1 using part (d) and Theorem 1-4, page 6. 4 4 (f) Since 10 and S are not mutually exclusive, we have, from Theorem 1-6, P(10 < S) P(10) P(S) P(10 > S) 1 13 1 4 1 52 4 13 (4 < C)r.

(a) Compute the voltage V for the coupled circuit shown in Fig. 14-24. polarity of one coil reversed.

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whose values can be represented by g(x1, . . . , xn ). The word statistic is often used for the random variable or for its values, the particular sense being clear from the context. In general, corresponding to each population parameter there will be a statistic to be computed from the sample. Usually the method for obtaining this statistic from the sample is similar to that for obtaining the parameter from a finite population, since a sample consists of a finite set of values. As we shall see, however, this may not always produce the best estimate, and one of the important problems of sampling theory is to decide how to form the proper sample statistic that will best estimate a given population parameter. Such problems are considered in later chapters. Where possible we shall try to use Greek letters, such as and , for values of population parameters, and Roman letters, m, s, etc., for values of corresponding sample statistics.

Coupling is a measure of how closely tied are two or more modules or classes. In particular, a coupling metric should indicate how likely would it be that a change to another module would a ect this module. Many coupling metrics have been proposed. The basic form of a coupling metric is to establish a list of items that cause one module to be tied to the internal workings of another module.

As we have seen, a sample statistic that is computed from X1, . . . , Xn is a function of these random variables and is therefore itself a random variable. The probability distribution of a sample statistic is often called the sampling distribution of the statistic. Alternatively we can consider all possible samples of size n that can be drawn from the population, and for each sample we compute the statistic. In this manner we obtain the distribution of the statistic, which is its sampling distribution. For a sampling distribution, we can of course compute a mean, variance, standard deviation, moments, etc. The standard deviation is sometimes also called the standard error.

CHAP. 14]

Let X1, X2, . . . , Xn denote the independent, identically distributed, random variables for a random sample of size n as described above. Then the mean of the sample or sample mean is a random variable defined by # X X1 X2 n c Xn (2)

Fig. 14-24 p (a) XM 0:8 5 10 5:66 , and so the Z-matrix is 3 j1 3 j1:66 Z 3 j1:66 8 j6 3 j1 50 3 j1:66 0 8:62 24:798 A I2 Z

in analogy with (3), page 75. If x1, x2, . . . , xn denote values obtained in a particular sample of size n, then the mean for that sample is denoted by x #

If a sample of size 5 results in the sample values 7, 9, 1, 6, 2, then the sample mean is x # 7 9 1 5 6 2 5

Then,

Let f (x) be the probability distribution of some given population from which we draw a sample of size n. Then # it is natural to look for the probability distribution of the sample statistic X, which is called the sampling distribution for the sample mean, or the sampling distribution of means. The following theorems are important in this connection. Theorem 5-1 The mean of the sampling distribution of means, denoted by mX , is given by # E(X) where is the mean of the population. mX m (4)

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