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W. Stevens, G. Myers, and L. Constantine, Structured Design, IBM Systems Journal, 13 #2, 1974, 115 139.

(p>2)

CHAP. 13]

t2 1

(p>2)

Ans:

(a) Compare the entries in the last row of the F0.95 table in Appendix F (corresponding to v2 entries under x2 in Appendix E. Then we see that 0.95 3.84 3.84 , 1 3.00 5.99 , 2 2.60 7.81 , 3 2.37 9.49 , 4 2.21 11.1 , 5

A 1 10 9 !2 j10 9 !, B 10 3 1 j! , C 10 6 j!, and D 1. At ! 1 rad/s, A 1, B 10 3 1 j , C 10 6 j, and D 1. At ! 103 rad/s, A % 1, B % j, C 10 3 j, and D 1. At ! 106 rad/s, A % 103 , B % 103 j, C j, and D 1

which provides the required verification. (b) Compare the entries in the last row of the F0.99 table in Appendix F (corresponding to v2 entries under x2 in Appendix E. Then we see that 0.99 6.63 6.63 , 1 4.61 9.21 , 2 3.78 11.3 , 3 3.32 13.3 , 4 3.02 15.1 , 5

Fig. 9-4

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which provides the required verification. The general proof of Theorem 4-10 follows by letting v2 S ` in the F distribution on page 116.

Fig. 13-36 13.35 A two-port network contains resistors, capacitors, and inductors only. With port #2 open [Fig. 13-37(a)], a unit step voltage v1 u t produces i1 e t u t mA and v2 1 e t u t (V). With port #2 shortcircuited [Fig. 13-37(b)], a unit step voltage v1 u t delivers a current i1 0:5 1 e 2t u t mA . Find i2 and the T-equivalent network. Ans: i2 0:5 1 e 2t u t [see Fig. 13-37(c)]

The bivariate normal distribution 4.51. Suppose that X and Y are random variables whose joint density function is the bivariate normal distribution. Show that X and Y are independent if and only if their correlation coefficient is zero.

Fig. 13-37 13.36 The two-port network N in Fig. 13-38 is speci ed by Z11 2, Z12 Z21 1, and Z22 4. Find I1 , I2 , and I3 . Ans: I1 24 A; I2 1:5 A; and I3 6:5 A

If the correlation coefficient r f (x, y) 0, then the bivariate normal density function (49), page 117, becomes c 1 e s1 !2p

and since this is a product of a function of x alone and a function of y alone for all values of x and y, it follows that X and Y are independent. Conversely, if X and Y are independent, f (x, y) given by (49) must for all values of x and y be the product of a function of x alone and a function of y alone. This is possible only if r 0.

Fig. 13-38

Miscellaneous distributions 4.52. Find the probability that in successive tosses of a fair die, a 3 will come up for the first time on the fifth toss.

14.1 MUTUAL INDUCTANCE The total magnetic ux linkage in a linear inductor made of a coil is proportional to the current passing through it; that is, Li (see Fig. 14-1). By Faraday s law, the voltage across the inductor is equal to the time derivative of the total in ux linkage; that is, v d di L dt dt

The version in Fig. 9.5 does not have good abstraction. That is, the details of the lower-level classes have not been hidden from the borrow function in library.

Method 1 The probability of not getting a 3 on the first toss is 5>6. Similarly, the probability of not getting a 3 on the second toss is 5>6, etc. Then the probability of not getting a 3 on the first 4 tosses is (5>6) (5>6) (5>6) (5>6) (5/6)4. Therefore, since the probability of getting a 3 on the fifth toss is 1>6, the required probability is 5 4 1 a b a b 6 6 625 7776

625 7776

Fig. 14-1

5, we see that the required proba-

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