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The theorem is also true under more general conditions; for example, it holds when X1, X2, . . . , Xn are independent random variables with the same mean and the same variance but not necessarily identically distributed.

CHAP. 11]

Suppose that events A1, A2, . . . , Ak are mutually exclusive, and can occur with respective probabilities p1, p2, . . . , pk where p1 p2 c pk 1. If X1, X2, . . . , Xk are the random variables respectively giving the number c X n, then of times that A1, A2, . . . , Ak occur in a total of n trials, so that X1 X2 k P(X1 n1, X2 n2, c, Xk nk) n pn1pn2 c pnk k n1!n2! c nk! 1 k (16)

where n1 n2 c nk n, is the joint probability function for the random variables X1, c, Xk. This distribution, which is a generalization of the binomial distribution, is called the multinomial distribution c p )n. since (16) is the general term in the multinomial expansion of ( p1 p2 k

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It should be noted that if the three impedances of one connection are equal, so are those of the equivalent connection, with Z =ZY 3.

Let A and B be two events (Fig. 1-3) such that P(A) 0. Denote by P(B u A) the probability of B given that A has occurred. Since A is known to have occurred, it becomes the new sample space replacing the original S. From this we are led to the definition P(B u A) ; or P(A> B) P(A) (17) (18)

Figure 11-13(a) shows a balanced Y-connected load. In many cases, for instance, in power calculations, only the common magnitude, IL , of the three line currents is needed. This may be obtained from the single-line equivalent, Fig. 11-13(b), which represents one phase of the original system, with the line-to-neutral voltage arbitrarily given a zero phase angle. This makes IL IL , where is the impedance angle. If the actual line currents IA , IB , and IC are desired, their phase angles may be found by adding to the phase angles of VAN , VBN , and VCN as given in Fig. 11-7. Observe that the angle on IL gives the power factor for each phase, pf cos . The method may be applied to a balanced -connected load if the load is replaced by its Yequivalent, where ZY 1 Z (Section 11.8). 3

P(A > B) ; P(A) P(B u A)

Fig. 1-3

Fig. 11-13 EXAMPLE 11.4 Rework Example 11.3 by the single-line equivalent method. Referring to Fig. 11-14 (in which the symbol Y indicates the type of connection of the original load), IL VLN 98:0 08 4:90 308 A Z 20 308 Hence,

Fig. 8-4.

In words, (18) says that the probability that both A and B occur is equal to the probability that A occurs times the probability that B occurs given that A has occurred. We call P(B u A) the conditional probability of B given A, i.e., the probability that B will occur given that A has occurred. It is easy to show that conditional probability satisfies the axioms on page 5.

From Fig. 11-7(b), the phase angles of VAN , VBN , and VCN are 908, 308, and 1508. IA 4:90 608 A IB 4:90 608 A

EXAMPLE 1.13 Find the probability that a single toss of a die will result in a number less than 4 if (a) no other information is given and (b) it is given that the toss resulted in an odd number.

The solution of the unbalanced delta-connected load consists in computing the phase currents and then applying KCL to obtain the line currents. The currents will be unequal and will not have the symmetry of the balanced case.

(a) Let B denote the event {less than 4}. Since B is the union of the events 1, 2, or 3 turning up, we see by Theorem 1-5 that P(B) P(1) P(2) P(3) 1 6 1 6 1 6 1 2

assuming equal probabilities for the sample points. (b) Letting A be the event {odd number}, we see that P(A) P(B u A)

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