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The Z-parameters of a two-port network are given by Z11 2s 1=s Z12 Z21 2s Z22 2s 4 Find I1 , I2 , V1 , and V2 .

(38)

is called chi square. Then we can show that for x P(x2 x)

The network is connected to a source and a load as shown in Fig. 13-8.

1 u(v>2) 1 e 2v>2 (v>2) 30

Fig. 13-8 The terminal characteristics are given by V1 2s 1=s I1 2sI2 V2 2sI1 2s 4 I2 17

(39)

Draw a system diagram for the simple vi-like editor. See Fig. 8-6.

and P( 2 x) 0 for x 0. The distribution defined by (39) is called the chi-square distribution, and v is called the number of degrees of freedom. The distribution defined by (39) has corresponding density function given by 1 x(v>2) 1 e 2v>2 (v>2) u0

The phasor representation of voltage vs t is Vs 12 V with s j. From KVL around the input and output loops we obtain the two additional equations (18) Vs 3I1 V1 0 1 s I2 V2 Substituting s j and Vs 12 in (17) and in (18) we get V1 jI1 2jI2 V2 2jI1 4 2j I2 12 3I1 V1 0 1 j I2 V2 from which I1 3:29 10:28 V1 2:88 37:58 I2 1:13 131:28 V2 1:6 93:88 18

0 (40) 0 v>2 2. (41)

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The Y-parameters may be obtained from the Z-parameters by solving (1) for I1 and I2 . Applying Cramer s rule to (1), we get Z22 Z V1 12 V2 DZZ DZZ Z21 Z I2 V 11 V DZZ 1 DZZ 2 I1

f (x)

It is seen that the chi-square distribution is a special case of the gamma distribution with Therefore, v,

19

M(t)

Fig. 8-6.

where DZZ Z11 Z22 Z12 Z21 is the determinant of the coe cients in (1). By comparing (19) with (9) we have

v>2,

2i )

20

For large v(v 30), we can show that !2x2 !2v 1 is very nearly normally distributed with mean 0 and variance 1. Three theorems that will be useful in later work are as follows: Theorem 4-3 Let X1, X2, . . . , Xv be independent normally distributed random variables with mean 0 and varic X 2 is chi-square distributed with v degrees of freedom. ance 1. Then x2 X 2 X 2 1 2 v Theorem 4-4 Let U1, U2, . . . , Uk be independent random variables that are chi-square distributed with v1, c U is chiv2, . . . , vk degrees of freedom, respectively. Then their sum W U1 U2 k c v degrees of freedom. square distributed with v1 v2 k Theorem 4-5 Let V1 and V2 be independent random variables. Suppose that V1 is chi-square distributed with v1 degrees of freedom while V V1 V2 is chi-square distributed with v degrees of freedom, where v v1. Then V2 is chi-square distributed with v v1 degrees of freedom. In connection with the chi-square distribution, the t distribution (below), the F distribution (page 116), and others, it is common in statistical work to use the same symbol for both the random variable and a value of that random variable. Therefore, percentile values of the chi-square distribution for v degrees of freedom are denoted by x2 , or briefly x2 if v is understood, and not by xp,v or xp. (See Appendix E.) This is an amp,v p biguous notation, and the reader should use care with it, especially when changing variables in density functions.

Given the Z-parameters, for the Y-parameters to exist, the determinant DZZ must be nonzero. versely, given the Y-parameters, the Z-parameters are Z11 Z12 Z21 Z22 Y22 DYY Y12 DYY Y21 DYY Y11 DYY

(42)

it is said to have Student s t distribution, briefly the t distribution, with v degrees of freedom. If v is large (v 30), the graph of f (t) closely approximates the standard normal curve as indicated in Fig. 4-2. Percentile

Con-

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