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We can take random samples from the population and then use these samples to obtain values that serve to estimate and test hypotheses about the population parameters. By way of illustration, let us consider Example 5.1 where X is a random variable whose values are the various heights. To obtain a sample of size 100, we must first choose one individual at random from the population. This individual can have any one value, say, x1, of the various possible heights, and we can call x1 the value of a random variable X1, where the subscript 1 is used since it corresponds to the first individual chosen. Similarly, we choose the second individual for the sample, who can have any one of the values x2 of the possible heights, and x2 can be taken as the value of a random variable X2. We can continue this process up to X100 since the sample size is 100. For simplicity let us assume that the sampling is with replacement so that the same individual could conceivably be chosen more than once. In this case, since the sample size is much smaller than the population size, sampling without replacement would give practically the same results as sampling with replacement. In the general case a sample of size n would be described by the values x1, x2, . . . , xn of the random variables X1, X2, . . . , Xn. In the case of sampling with replacement, X1, X2, . . . , Xn would be independent, identically distributed random variables having probability distribution f(x). Their joint distribution would then be P(X1 x1, X2 x2, c, Xn xn) f (x1) f (x2) c f (xn) (1)

EXAMPLE 14.8 Given L1 0:2 H, L2 0:1 H, M 0:1 H, and R 10 in the circuit of Fig. 14-17. for v1 142:3 sin 100t. Find i1

Any quantity obtained from a sample for the purpose of estimating a population parameter is called a sample statistic, or briefly statistic. Mathematically, a sample statistic for a sample of size n can be defined as a function of the random variables X1, . . . , Xn, i.e., g(X1, . . . , Xn ). The function g(X1, . . . , Xn ) is another random variable,

Fig. 14-17 The input impedance Z1 at ! 100 is [see (18)] Z1 Then, or p V1 M 2 !2 0:01 10 000 j!L1 j20 5 j15 5 10 71:68 I1 Z2 j!L2 10 j10 I1 V1 =Z1 9 71:68 A i1 9 sin 100t 71:68 A

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Random experiments, sample spaces, and events 1.1. A card is drawn at random from an ordinary deck of 52 playing cards. Describe the sample space if consideration of suits (a) is not, (b) is, taken into account.

(a) If we do not take into account the suits, the sample space consists of ace, two, . . . , ten, jack, queen, king, and it can be indicated as {1, 2, . . . , 13}. (b) If we do take into account the suits, the sample space consists of ace of hearts, spades, diamonds, and clubs; . . . ; king of hearts, spades, diamonds, and clubs. Denoting hearts, spades, diamonds, and clubs, respectively, by 1, 2, 3, 4, for example, we can indicate a jack of spades by (11, 2). The sample space then consists of the 52 points shown in Fig. 1-5.

EXAMPLE 14.9 Referring to Example 14.8, let v1 u t . The input impedance is [see (17)] Z1 s L1 s Substituting the given values for the elements, we get Z1 s s s 200 10 s 100 or

Fig. 1-5

For t > 0, the input v1 1 V is an exponential est whose exponent s 0 is a pole of Y1 s . Therefore, i1;f kt with k 1=L1 5. This result may also be obtained directly by dc analysis of the circuit in Fig. 14-17.

Fig. 9-7.

1.2. Referring to the experiment of Problem 1.1, let A be the event {king is drawn} or simply {king} and B the event {club is drawn} or simply {club}. Describe the events (a) A < B, (b) A > B, (c) A < Br, (d) Ar < Br, (e) A B, (f ) Ar Br, (g) (A > B) < (A > Br).

(a) A < B (b) A > B (c) Since B {either king or club (or both, i.e., king of clubs)}. {both king and club} {club}, Br {king of clubs}. {heart, diamond, spade}.

14.1 When one coil of a magnetically coupled pair has a current 5.0 A the resulting uxes 11 and 12 are 0.2 mWb and 0.4 mWb, respectively. If the turns are N1 500 and N2 1500, nd L1 , L2 , M, and the coe cient of coupling k.

{not club}

L1

{king or heart or diamond or spade}. {not king of clubs} {any card but king of clubs}.

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