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In other words, in the long run it becomes extremely likely that the proportion of successes, X>n, will be as close as you like to the probability of success in a single trial, p. This law in a sense justifies use of the empirical definition of probability on page 5. A stronger result is provided by the strong law of large numbers (page 83), which states that with probability one, lim X>n p, i.e., X>n actually converges to p except in a negligible nS` number of cases.

Fig. 8-2.

The generator coil ends can be connected as shown in Fig. 11-6, making a delta-connected (or connected), three-phase system with lines a, b, and c. A delta-connected set of coils has no neutral point to produce a four-wire system, except through the use of -Y transformers.

One of the most important examples of a continuous probability distribution is the normal distribution, sometimes called the Gaussian distribution. The density function for this distribution is given by 1 (4) e (x m)2/2s2 ` x ` s22p are the mean and standard deviation, respectively. The corresponding distribution function is f (x) F(x) P(X x)

If X has the distribution function given by (5), we say that the random variable X is normally distributed with mean and variance 2. If we let Z be the standardized variable corresponding to X, i.e., if we let Z X s m (6)

Fig. 11-6

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then the mean or expected value of Z is 0 and the variance is 1. In such cases the density function for Z can be 0 and 1, yielding obtained from (4) by formally placing 1 (7) e z2>2 22p This is often referred to as the standard normal density function. The corresponding distribution function is given by f (z)

The selection of a phase angle for one voltage in a three-phase system xes the angles of all other voltages. This is tantamount to xing the t 0 point on the horizontal axis of Fig. 11-3, which can be done quite arbitrarily. In this chapter, an angle of zero will always be associated with the phasor voltage of line B with respect to line C: VBC VL 08. p It is shown in Problem 11.4 that the line-to-line voltage VL is 3 times the line-to-neutral voltage. All ABC-sequence voltages are shown in Fig. 11-7(a) and CBA voltages in Fig. 11-7(b). These phasor

z z 1 1 1 u2>2 du u2>2 du (8) 3 `e 30 e 2 !2p !2p We sometimes call the value z of the standardized variable Z the standard score. The function F(z) is related to the extensively tabulated error function, erf(z). We have

F(z)

Fig. 11-7

Although not used much in OO development, data ow diagrams (see Section 2.2) were essential parts of pre-OO software development. Data ow diagrams (DFDs) still have an important role in the speci cation of many systems. The importance of data ow diagrams is in specifying what data is available to a component. Knowing the data available often helps in the understanding of what a component is expected to do and how it will accomplish the task.

erf(z)

voltages, in keeping with the previous chapters, re ect maximum values. In the three-phase, four-wire, 480-volt system, widely used for industrial loads, and the 208-volt system, common in commercial buildings, e ective values are speci ed. In this chapter, a line-to-line voltage in the former system p would be VBC 678:8 08 V, making VBC eff 678:8= 2 480 V. People who regularly work in this eld use e ective-valued phasors, and would write VBC 480 08 V.

F(z)

Three identical impedances connected as shown in Fig. 11-8 make up a balanced -connected load. The currents in the impedances are referred to either as phase currents or load currents, and the three will be equal in magnitude and mutually displaced in phase by 1208. The line currents will also be equal in magnitude and displaced from one another by 1208; by convention, they are given a direction from the source to the load.

1 c1 2

z bd !2

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