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3.92. Find (a) the mode, (b) the median of a random variable X having density function f (x) and (c) compare with the mean. 3.93. Work Problem 3.100 if the density function is f (x) e 4x(1 0 x2) 0 x 1 otherwise e e 0

where Vp and Ip are the rms values of v and i, respectively, and is the angle between them. The power oscillates between Vp Ip 1 cos and Vp Ip 1 cos . In power systems, especially at higher levels, it is desirable to have a steady ow of power from source to load. For this reason, polyphase systems are used. Another advantage is having more than one voltage value on the lines. In polyphase systems, Vp and Ip indicate voltage and current, respectively, in a phase which may be di erent from voltages and currents in other phases. This chapter deals mainly with three-phase circuits which are the industry standard. However, examples of two-phase circuits will also be presented.

3.94. Find (a) the median, (b) the mode for a random variable X defined by X e 2 1 prob. 1>3 prob. 2>3

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A balanced two-phase generator has two voltage sources producing the same amplitude and frequency but 908 or 1808 out of phase. There are advantages in such a system since it gives the user the option of two voltages and two magnetic elds. Power ow may be constant or pulsating.

3.95. Find (a) the median, (b) the mode of the set of numbers 1, 3, 2, 1, 5, 6, 3, 3, and (c) compare with the mean.

3.96. Find the (a) 25th, (b) 75th percentile values for the random variable having density function f (x) e 2(1 0 x) x 1 0 otherwise

EXAMPLE 11.1 An ac generator contains two voltage sources with voltages of the same amplitude and frequency, but 908 out of phase. The references of the sources are connected together to form the generator s reference terminal n. The system feeds two identical loads [Fig. 11-1(a)]. Find currents, voltages, the instantaneous and average powers delivered. Terminal voltages and currents at generator s terminal are p p vb t Vp 2 cos !t 908 2 va t Vp 2 cos !t p p ia t Ip 2 cos !t ib t Ip 2 cos !t 908 In the phasor domain, let Z jZj and Ip Vp =jZj. VAN Vp 0 IA Ip VBN Vp 908 IB Ip 908 Then, p VAB VAN VBN 2Vp 458 p IN IA IB Ip 2 458 3

Software Requirements Speci cation Software Design Using UML and OCL Software Test Plan Software Test Report User Manual Each document will be reviewed in the draft version and inspected in the nal version. The SQA lead (John) will be responsible for conducting all reviews and inspections.

3.97. Find the (a) 10th, (b) 25th, (c) 75th, (d) 90th percentile values for the random variable having density function f (x) e c(x 0 x3) 0 x 1 otherwise

The voltage and current phasors are shown in Fig. 11-1(b). Instantaneous powers pA t and pB t delivered by the two sources are

3.98. Find (a) the semi-interquartile range, (b) the mean deviation for the random variable of Problem 3.96. 3.99. Work Problem 3.98 for the random variable of Problem 3.97.

(a) f(x)

In Fig. 2-11 the plots of i and v are given. Since the maximum current is 5.0 A, the maximum stored energy is Wmax 1 2 25:0 mJ LI 2 max

(b) f (x)

CHAP. 2]

1 p(1 x2)

XYZ Corporation Coding Standards will be used. The SQA team will conduct inspections of all code to ensure compliance. The compliance report will be submitted to the project lead. Unadjusted function points will be counted on all components. LOC will be counted on all classes.

3.101. Obtain the probability that the random variable X differs from its mean by more than the semi-interquartile range in the case of (a) Problem 3.96, (b) Problem 3.100(a).

Fig. 2-10

3.102. Find the coefficient of (a) skewness, (b) kurtosis for the distribution of Problem 3.100(a). 3.103. If f (x) c Q1 0 u xu a R u xu u xu a a

Fig. 2-11

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