ANSWERS TO THE EXERCISES 12/8 



11/11. 1 : Yes. D has a variety of 10 bits/sec, the optic nerve can transmit 200 

 times this. 2: The capacity available for regulation is 0-63 bits/sec 

 by telegraph and 5-64 bits/sec by the wheel. So evidently D does not 

 usually emit more than 6-3 bits/sec. 3: No, it is grossly insufficient. 

 D provides lO"? bits in each day, and the variety transmitted to the general 

 is at most one-seventeenth of this. 4: No, he can emit only 3-6 x 105 

 bits/day. 



11/14. 1: 



2: D is threatening to transmit to £ at 2 bits/sec. To reduce this to 

 zero the channel D -^ R must transmit at not less than this rate. 3: 

 C -> £■ is to carry 20 bits/sec, therefore C -^ R must carry at least that 

 amount. A: R-> Tmust carry 2 bits/sec to neutralise D (from Ex. 2), 

 and 20 bits/sec from C; as these two are independent (D's values and 

 C's not correlated), the capacity must be at least 22 bits/sec. 



12/8. 1: 



2: The systems are almost isomorphic; jS, however, will occasionally 

 jump from A directly to D, and will occasionally stay at C for a step. 

 3: The successive probabilities for a at each step are: 0, |, iV and f|; 



6's probabilities are the remainder 

 pre-multiplying the column vector 



ra 



4: The answer can be found by 

 by the matrix product prq; com- 



pare Ex. 2/16/3 and 12/8/4. 5: The new system must have states that 

 are couples, e.g. {b,e); so it will have six states. Now find the transition 

 probabilities. What, for instance, is that for the transition {b,e) -^ 

 (a J) ? For this to occur, b must go to a, and it must do this while the 

 other component is at e, i.e. at p. With input at /3 the probability of 

 b-^a is 0-9. Similarly, with b (i.e. S) the probability of e -^fis 0-3; 

 so the probability of the whole transition (for which both independent 

 events must occur) is 0-27. The other probabilities can be found 

 similarly, and the matrix is (with brackets omitted for brevity) : 



6: Yes. 



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