INCESSANT TRANSMISSION 9/6 



The equilibria! values of a Markov chain are readily computed. 

 At equilibrium the values are unchanging, so dg\ say, is equal to 

 dg. So the first line of the equation becomes 



^B =^ 4"b + ^.^w + 8"P 

 i.e. 0= -1^5+ t^p^+ i^/> 



The other lines are treated similarly. The lines are not all independ- 

 ent, however, for the three populations must, in this example, sum 

 to 100; one line (any one) is therefore struck out and replaced by 



dB+ d^-^ dp= 100. 



The equations then become, e.g., 



-¥B+ld^-^yp= 



ds + dyy -\- dp= 100 

 -1/^ — If] — 



which can be solved in the usual way. In this example the equilibrial 

 values are (44-9, 42-9, 12-2); as S.9/5 predicted, any individual 

 insect does not spend much time under the pebbles. 



Ex. 1 : Find the populations that would follow the initial state of putting all 



the insects on the bank. 

 Ex. 2: Verify the equilibrial values. 

 Ex. 3 : A six-sided die was heavily biased by a weight hidden in face .y. When 



placed in a box with face / upwards and given a thorough shaking, the 



probability that it would change to face g was found, over prolonged 



testing, to be: 



\ 



1 



2 

 3 

 g 4 

 5 

 6 



Which is X ? (Hint : Beware !) 



Ex.A:K compound AB is dissolved in water. In each small interval of time 

 each molecule has a 1% chance of dissociating, and each dissociated A 

 has an 0-1% chance of becoming combined again. What is the matrix 

 of transition probabilities of a molecule, the two states being "dissociated" 

 and "not dissociated"? (Hint: Can the number of 5's dissociated be 

 ignored ?) 



Ex. 5 : (Continued.) What is the equilibrial value of the percentage dissociated? 



Ex. 6: Write out the transformations of (i) the individual insect's transitions, 

 and (ii) the population's transitions. How are they related? 



Ex. 7: How many states appear in the insect's transitions? How many in the 

 system of populations ? 



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