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EXAMPLE 9.6 Suppose that the phasor voltage across ZB , with polarity as indicated in Fig. 9-13 is sought. Choosing meshes as in Fig. 9-12 would entail solving for both I1 and I2 , then obtaining the voltage as VB I2 I1 ZB . In Fig. 9-13 three loops (two of which are meshes) are chosen so as to make I1 the only current in ZB . Furthermore, the direction of I1 is chosen such that VB I1 ZB . Setting up the matrix equation:

2.50. Find the distribution function for the random variable X of Problem 2.49. 2.51. The distribution function of a random variable X is given by F(x) If P(X 3) cx3 1 0 0 x x x 3 3 0 1), (d) P(1 X 2).

f 400

2.52. Can the function F(x) be a distribution function Explain. 2.53. Let X be a random variable having density function f (x) Find (a) the value of the constant c, (b) P( 2

Fig. 9-13

c(1 0

Input and Transfer Impedances The notions of input resistance (Section 4.5) and transfer resistance (Section 4.6) have their exact counterparts in the frequency domain. Thus, for the single-source network of Fig. 9-14, the input impedance is Zinput;r Vr z Ir rr

3 2 ),

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2.54. The joint probability function of two discrete random variables X and Y is given by f(x, y) cxy for x 1, 2, 3 and y 1, 2, 3, and equals zero otherwise. Find (a) the constant c, (b) P(X 2, Y 3), (c) P(l X 2, Y 2), (d) P(X 2), (e) P(Y 2), (f) P(X 1), (g) P(Y 3). 2.55. Find the marginal probability functions of (a) X and (b) Y for the random variables of Problem 2.54. (c) Determine whether X and Y are independent. 2.56. Let X and Y be continuous random variables having joint density function f (x, y) Determine (a) the constant c, (b) P(X independent. e

where rr is the cofactor of Zrr in z ; and the transfer impedance between mesh (or loop) r and mesh (loop) s is Ztransfer;rs where rs is the cofactor of Zrs in z . V r z Is rs

1 2 ),

g 144

Fig. 9-14

y 2) (c) P ( 1 4

3 4 ),

As before, the superposition principle for an arbitrary n-mesh or n-loop network may be expressed as V1 Ztransfer;1k Vk 1 Ztransfer; k 1 k Vk Zinput;k Vk 1 Ztransfer; k 1 k Vn Ztransfer;nk

1 2 ),

CHAP. 9]

2.57. Find the marginal distribution functions (a) of X and (b) of Y for the density function of Problem 2.56.

2.58. Find the conditional probability function (a) of X given Y, (b) of Y given X, for the distribution of Problem 2.54. 2.59. Let f (x, y) e x 0 y x 1, 0 0 otherwise y 1

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