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The central limit theorem 4.24. Verify the central limit theorem for a random variable X that is binomially distributed, and thereby establish the validity of the normal approximation to the binomial distribution.

Parallel Connection Figure 13-13 shows a parallel connection of two-port networks a and b with short-circuit admittance parameters Ya and Yb . In this case, the Y-parameters are convenient to work with. The Y-parameters of the parallel connection are (see Problem 13.11): Y11 Y11;a Y11;b Y12 Y12;a Y12;b Y21 Y21;a Y21;b Y22 Y22;a Y22;b or, in the matrix form Y Ya Yb 32b

np)>1npq)

32a

The standardized variable for X is X* E(etX*)

Fig. 13-13

E(etX>1npq )

Estimated Completion Date 1/25/01 2/15/01 5/15/01 4/15/01 7/1/01 9/01/01

u3 3!

Cascade Connection The cascade connection of two-port networks a and b is shown in Fig. 13-14. In this case the T-parameters are particularly convenient. The T-parameters of the cascade combination are

Therefore,

or, in the matrix form,

E(etX*)

But as n S ` , the right-hand side approaches et2/2, which is the moment generating function for the standard normal distribution. Therefore, the required result follows by Theorem 3-10, page 77.

33b

4.25. Prove the central limit theorem (Theorem 4-2, page 112).

Actual Date Completed 2/1/01 2/15/01

Fig. 13-14

For n Thus 1, 2, . . . , we have Sn X1 X2 E(X1) c Xn. Now X1, X2, . . . , Xn each have mean and variance s2. c E(Xn) nm

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E(Sn) and, because the Xk are independent, Var (Sn)

What types of parameters are appropriate to and can best describe a given two-port network or device Several factors in uence the choice of parameters. (1) It is possible that some types of parameters do not exist as they may not be de ned at all (see Example 13.10). (2) Some parameters are more convenient to work with when the network is connected to other networks, as shown in Section 13.11. In this regard, by converting the two-port network to its T- and Pi-equivalent and then applying the familiar analysis techniques, such as element reduction and current division, we can greatly reduce and simplify the overall circuit. (3) For some networks or devices, a certain type of parameter produces better computational accuracy and better sensitivity when used within the interconnected circuit.

E(X2)

EXAMPLE 13.10 Find the Z- and Y-parameters of Fig. 13-15.

Var (X1)

Var (X2)

Fig. 13-15 We apply KVL to the input and output loops. Thus, Input loop: Output loop: or By comparing (34) and (2) we get Z11 6 Z12 3 Z21 10 Z22 5 V1 3I1 3 I1 I2 V2 7I1 2I2 3 I1 I2 V1 6I1 3I2 and V2 10I1 5I2 (34)

4/1/01

Var (Xn)

The Y-parameters are, however, not de ned, since the application of the direct method of (10) or the conversion from Z-parameters (19) produces DZZ 6 5 3 10 0.

It follows that the standardized random variable corresponding to Sn is S* n The moment generating function for

* E(etSn)

Terminal parameters are de ned by the following equations Z-parameters V1 Z11 I1 Z12 I2 V2 Z21 I1 Z22 I2 V Z I Y-parameters I1 Y11 V1 Y12 V2 I2 Y21 V1 Y22 V2 I Y V h-parameters V1 h11 I1 h12 V2 I2 h21 I1 h22 V2 T-parameters V1 AV2 BI2 I1 CV2 DI2

nm)>s1n] m)>s1net(X2 m)>s1n m)>s1n]

CHAP. 13]

m)>s1n] m)>s1n]

m)>s1n]

The BAC is the sum of the estimations. BAC = 330 days. BAC is an estimate of the total work. On 4/1/01, tasks 1,2, and 4 have been completed. The BCWP is the sum of the BCWS for those tasks. So BCWP is 70 days. The earned value (EV) is 70/330, or 21.2 percent. On 4/1/01 tasks 1 and 2 were scheduled to be completed and 1,2, and 4 were actually completed. So BCWP is 70 days and BCWS is 30 days. Thus, SPI is 70/30, or 233 percent. The SV = 70 days 30 days = +40 days, or 40 programmer-days ahead. The ACWP is the sum of actual effort for tasks 1, 2, and 4. So, ACWP is 80 programmer-days. CPI is 70/80 = 87.5 percent. The CV = 70 programmer-days 80 programmer-days = 10 programmer-days, or 10 programmer-days behind.

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