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where is a given positive constant. This distribution is called the Poisson distribution (after S. D. Poisson, who discovered it in the early part of the nineteenth century), and a random variable having this distribution is said to be Poisson distributed. The values of f (x) in (13) can be obtained by using Appendix G, which gives values of e for various values of .

The voltage-current phasor diagram is shown in Fig. 11-11. Note that with one line current calculated, the other two can be obtained through the symmetry of the phasor diagram. All three line currents return through the neutral. Therefore, the neutral current is the negative sum of the line currents:

Some important properties of the Poisson distribution are listed in Table 4-3.

Table 4-3

Fig. 11-11

Mean Variance Standard deviation Coefficient of skewness Coefficient of kurtosis Moment generating function Characteristic function s a3 a4 M(t) f(v) 3

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Since the neutral current of a balanced, Y-connected, three-phase load is always zero, the neutral conductor may, for computation purposes, be removed, with no change in the results. In actual power circuits, it must not be physically removed, since it carries the (small) unbalance of the currents, carries short-circuit or fault currents for operation of protective devices, and prevents overvoltages on the phases of the load. Since the computation in Example 11.3 proceeded without di culty, the neutral will be included when calculating line currents in balanced loads, even when the system is actually threewire.

The use case diagram represents the functionality of the system from the user s point of view (see Section 2.5). All critical functionality must be mentioned. However, routine functions that are implied by a higher-level phrase do not have to be speci cally mentioned (the danger of miscommunication must be balanced by clarity). The textual requirements will detail these individual functions.

1) 1)

In the binomial distribution (1), if n is large while the probability p of occurrence of an event is close to zero, so that q 1 p is close to 1, the event is called a rare event. In practice we shall consider an event as rare if the number of trials is at least 50 (n 50) while np is less than 5. For such cases the binomial distribution is very closely approximated by the Poisson distribution (13) with np. This is to be expected on comparing Tables 4-1 and 4-3, since by placing np, q < 1, and p < 0 in Table 4-1, we get the results in Table 4-3.

Figure 11-12 shows three impedances connected in a (delta) con guration, and three impedances connected in a Y (wye) con guration. Let the terminals of the two connections be identi ed in pairs as indicated by the labels , ,

Since there is a relation between the binomial and normal distributions and between the binomial and Poisson distributions, we would expect that there should also be a relation between the Poisson and normal distributions. This is in fact the case. We can show that if X is the Poisson random variable of (13) and (X l)> !l is the corresponding standardized random variable, then

. Then Z1 is the impedance adjoining terminal in the Y-connection, and ZC is the impedance opposite terminal in the -connection, and so on. Looking into any two terminals, the two connections will be equivalent if corresponding input, output, and transfer impedances are equal. The criteria for equivalence are as follows:

The similarity between (12) and (14) naturally leads us to ask whether there are any other distributions besides the binomial and Poisson that have the normal distribution as the limiting case. The following remarkable theorem reveals that actually a large class of distributions have this property. Theorem 4-2 (Central Limit Theorem) Let X1, X2, . . . , Xn be independent random variables that are identically distributed (i.e., all have the same probability function in the discrete case or density function in the continuous case) and have finite mean and variance 2. Then if Sn X1 X2 . . . Xn (n l, 2 . . .),

(15)

Fig. 11-12

Draw the use case diagram for an editor that is similar to a simplified Unix vi-like editor. See Fig. 8-4.

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