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where 5 is the region shown shaded in Fig. 2-25. The right side of (2) is the area of this region, which is equal 3 7 to 1 (4 )( 3) 16, since the square has area 1, while the two corner triangles have areas 1 ( 3)(3 ) each. Thus the 4 2 4 4 required probability is 7 > 16.

Consider the frequency-domain network of Fig. 9-12. Applying KVL, as in Section 4.3, or simply by inspection, we nd the matrix equation

1 4

d 225

Fig. 2-25

CHAP. 9]

2.38. A coin is tossed three times. If X is a random variable giving the number of heads that arise, construct a table showing the probability distribution of X. 2.39. An urn holds 5 white and 3 black marbles. If 2 marbles are to be drawn at random without replacement and X denotes the number of white marbles, find the probability distribution for X. 2.40. Work Problem 2.39 if the marbles are to be drawn with replacement. 2.41. Let Z be a random variable giving the number of heads minus the number of tails in 2 tosses of a fair coin. Find the probability distribution of Z. Compare with the results of Examples 2.1 and 2.2. 2.42. Let X be a random variable giving the number of aces in a random draw of 4 cards from an ordinary deck of 52 cards. Construct a table showing the probability distribution of X.

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for the unknown mesh currents I1 ; I2 ; I3 . Here, Z11 ZA ZB , the self-impedance of mesh 1, is the sum of all impedances through which I1 passes. Similarly, Z22 ZB ZC ZD and Z33 ZD ZE are the self-impedances of meshes 2 and 3.

2.43. The probability function of a random variable X is shown in Table 2-7. Construct a table giving the distribution function of X. Table 2-7 x f (x) 1 1>2 2 1>3 3 1>6 x F(x) 1 1>8 Table 2-8 2 3>8 3 3>4 4 1

2.44. Obtain the distribution function for (a) Problem 2.38, (b) Problem 2.39, (c) Problem 2.40. 2.45. Obtain the distribution function for (a) Problem 2.41, (b) Problem 2.42. 2.46. Table 2-8 shows the distribution function of a random variable X. Determine (a) the probability function,

Fig. 9-12

(b) P(1

e 64

The 1,2-element of the Z-matrix is de ned as: X Z12 (impedance common to I1 and I2 where a summand takes the plus sign if the two currents pass through the impedance in the same direction, and takes the minus sign in the opposite case. It follows that, invariably, Z12 Z21 . In Fig. 9-12, I1 and I2 thread ZB in opposite directions, whence Z12 Z21 ZB Similarly, Z13 Z31 Z23 Z23 X X (impedance common to I1 and I3 0 (impedance common to I2 and I3 ZD

1.4).

The Z-matrix is symmetric. In the V-column on the right-hand side of the equation, the entries Vk (k 1; 2; 3) are de ned exactly as in Section 4.3: X (driving voltage in mesh k Vk where a summand takes the plus sign if the voltage drives in the direction of Ik , and takes the minus sign in the opposite case. For the network of Fig. 9-12, V1 Va V2 0 V3 Vb

0 0 1).

2.48. Find the distribution function for the random variable of Problem 2.47. Graph the density and distribution functions, describing the relationship between them. 2.49. A random variable X has density function f (x) Find (a) the constant c, (b) P(X 2), (c) P(1 > 2 X cx 2 cx 0 1 x 2 2 x 3 otherwise

Instead of using the meshes, or windows of the (planar) network, it is sometimes expedient to choose an appropriate set of loops, each containing one or more meshes in its interior. It is easy to see that two loop currents might have the same direction in one impedance and opposite directions in another. Nevertheless, the preceding rules for writing the Z-matrix and the V-column have been formulated in such a way as to apply either to meshes or to loops. These rules are, of course, identical to those used in Section 4.3 to write the R-matrix and V-column.

3 > 2).

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