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4.102. Find (a) the third, (b) the fourth moment about the mean of a uniform distribution. 4.103. Determine the coefficient of (a) skewness, (b) kurtosis of a uniform distribution. 4.104. If X and Y are independent and both uniformly distributed in the interval from 0 to 1, find P( Z X YZ

where the last step involved (8) and (10): q M k a2 L2 L2 kaL2 Now, de ning the magnetizing current i by the equation i1 we have e2 M di dt or E2 jXM I 13 i2 i a or I1 I2 I a 12

1 2 ).

According to (13), the magnetizing current may be considered to set up the mutual ux m in the core. In terms of coil emfs and leakage reactances, an equivalent circuit for the linear transformer may be drawn, in which the primary and secondary are e ectively decoupled. This is shown in Fig. 14-11(a); for comparison, the dotted equivalent circuit is shown in Fig. 14-11(b).

4.105. Suppose that X is Cauchy distributed according to (29), page 114, with a (b) P(X 2 12). 2. Find (a) P(X 2),

4.106. Prove that if X1 and X2 are independent and have the same Cauchy distribution, then their arithmetic mean also has this distribution. 4.107. Let X1 and X2 be independent and normally distributed with mean 0 and variance 1. Prove that Y Cauchy distributed. X1 > X2 is

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Fig. 14-11 EXAMPLE 14.5 Draw the voltage-current phasor diagram corresponding to Fig. 14-11(a), and from it derive the input impedance of the transformer. The diagram is given in Fig. 14-12, in which L denotes the phase angle of ZL . Note that, in accordance with (13), the induced emfs E1 and E2 lead the magnetizing current I by 908. The diagram yields the three phasor equations V1 ajXM I R1 jX11 I1 jXM I ZL R2 jX22 I2 1 I1 I2 I a

Draw a use case diagram for the simple library. (See Fig. 2-10.)

4.108. A random variable X is gamma distributed with 3, 2. Find (a) P(X 1), (b) P(l X 2).

Fig. 14-12 Elimination of I2 and I among these equations results in V1 jXM =a R2 jX22 ZL Zin R1 jX11 a2 I1 jXM =a R2 jX22 ZL If, instead, the mesh current equations for Fig. 14-11(b) are used to derive Zin , the result is Zin R1 jX1

4.109. For a chi-square distribution with 12 degrees of freedom, find the value of x2 such that (a) the area to the right c of x2 is 0.05, (b) the area to the left of x2 is 0.99, (c) the area to the right of x2 is 0.025. c c c 4.110. Find the values of x2 for which the area of the right-hand tail of the x2 distribution is 0.05, if the number of degrees of freedom v is equal to (a) 8, (b) 19, (c) 28, (d) 40. 4.111. Work Problem 4.110 if the area of the right-hand tail is 0.01.

4.112. (a) Find x2 and x2 such that the area under the x2 distribution corresponding to v 20 between x2 and x2 is 1 2 1 2 0.95, assuming equal areas to the right of x2 and left of x2. (b) Show that if the assumption of equal areas in 2 1 part (a) is not made, the values x2 and x2 are not unique. 1 2 4.113. If the variable U is chi-square distributed with v 7, find x2 and x2 such that (a) P(U 1 2 (b) P(U x2) 0.50, (c) P(x2 U x2) 0.90. 1 1 2 4.114. Find (a) x2 and (b) x2 for v 0.05 0.95 4.115. Find (a) x2 and (b) x2 for v 0.025 0.975 150. 250. x2) 2 0.025,

14a

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