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The Poisson distribution 4.20. Establish the validity of the Poisson approximation to the binomial distribution.

Fig. 13-10 This is the simple model of a eld e ect transistor in its linear region of operation. To nd the g-parameters, we rst derive the terminal equations by applying Kirchho s laws at the terminals: At the input terminal: At the output terminal: or I1 10 9 V1 V1 109 I1 V2 10 I2 10 3 V1 and V2 10I2 10 2 V1 (28)

If X is binomially distributed, then (1) where E(X) np. Let P(X np so that p x) P(X x) n a b p xq n x

2) c (n x!nx 2 c a1 nb x!

The transmission parameters A, B, C, and D express the required source variables V1 and I1 in terms of the existing destination variables V2 and I2 . They are called ABCD or T-parameters and are de ned by

Now as n S ` ,

9. Draw a state diagram for the B&B problem as a whole (see Problem 4.3). See Fig. 8-14.

29

S (e l)(1)

stays fixed (i.e., p S 0), P(X lxe x) S x!

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Fig. 13-11 This is the simple lumped model of an incremental segment of a transmission line. V1 Z Zb a 1 Za Yb V2 I2 0 Zb V B 1 Za I2 V2 0 I C 1 Yb V2 I2 0 I D 1 1 I A

Another method The moment generating function for the binomial distribution is (3) If (4) As n S ` this approaches (5) el(et

2 V2 0

which is the moment generating function of the Poisson distribution. The required result then follows on using Theorem 3-10, page 77.

First, we see that P(X x)

30

Company LMN is partway through its project. The job log below indicates the current status of the project.

0 for x x)

Two-port networks may be interconnected in various con gurations, such as series, parallel, or cascade connection, resulting in new two-port networks. For each con guration, certain set of parameters may be more useful than others to describe the network.

lx a x!

4.22. Ten percent of the tools produced in a certain manufacturing process turn out to be defective. Find the probability that in a sample of 10 tools chosen at random, exactly 2 will be defective, by using (a) the binomial distribution, (b) the Poisson approximation to the binomial distribution.

Series Connection Figure 13-12 shows a series connection of two two-port networks a and b with open-circuit impedance parameters Za and Zb , respectively. In this con guration, we use the Z-parameters since they are combined as a series connection of two impedances. The Z-parameters of the series connection are (see Problem 13.10):

(a) The probability of a defective tool is p 0.1. Let X denote the number of defective tools out of 10 chosen. Then, according to the binomial distribution, P(X (b) We have np P(X (10)(0.1) x) 2) a 10 b(0.1)2(0.9)8 2 0.1937 or 0.19

Fig. 13-12

4.23. If the probability that an individual will suffer a bad reaction from injection of a given serum is 0.001, determine the probability that out of 2000 individuals, (a) exactly 3, (b) more than 2, individuals will suffer a bad reaction.

Estimated E ort (programmerdays) 5 25 120 40 60 80

Let X denote the number of individuals suffering a bad reaction. X is Bernoulli distributed, but since bad reactions are assumed to be rare events, we can suppose that X is Poisson distributed, i.e., P(X (a) (b) P(X 2) x) lxe x!

CHAP. 13]

where 3) [P(X c 20e 0!

Z11 Z11;a Z11;b Z12 Z12;a Z12;b Z21 Z21;a Z21;b Z22 Z22;a Z22;b or, in the matrix form, Z Za Zb

(2000)(0.001) 0.180 1) 22e 2 d 2! P(X 2)]

23e 3! 0)

31a

P(X 21e 1!

31b

Actual E ort So Far (programmerdays) 10 20 80 50 50 70

An exact evaluation of the probabilities using the binomial distribution would require much more labor.

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