36 BELL SYSTEM TECHNICAL JOURNAL 



of) the deviations of certain internal parameters from their nominal 

 values. Another example is furnished by two telephone lines con- 

 nected by the usual type of two-way telephone repeater: If the 

 two lines and their associated apparatus could be made identically 

 alike, a state of perfect balance would exist at the repeater, and 

 there would be no tendency for the repeater to sing; however, as a 

 result of manufacturing variations, perfect balance is unattainable and 

 thus the practicable amplification obtainable from the repeater is 

 limited by the manufacturing deviations of the lines and associated 

 apparatus — particularly the deviations in the inductances and spac- 

 ings of the loading coils, in case the lines are loaded. 



Such examples may furnish at least the three types of probability 

 problems described in the following three paragraphs : 



Before the construction of the system there may arise the "direct" 

 problem of calculating the characteristic to be expected, corresponding 

 to the known (or assumed) ranges of the manufacturing variations in 

 the elements. Before the elements are manufactured, the deviation 

 of any element from its nominal value is of course unknown; more- 

 over, such deviation is not completely predictable, since from its very 

 nature it depends on chance. The deviation is a variable in the sense 

 that it can take any value within a certain possible range. But it is 

 a particular sort of variable, namely a chance-variable, in the sense 

 that there exists a certain chance or probability that the deviation 

 will lie within any stated range of values, with the chance depending 

 of course on this range and on the specific probability law of deviation 

 for the kind of element under consideration. Correspondingly the 

 deviation of the contemplated transmission characteristic of the pro- 

 posed system is a chance-variable, whose probability law depends of 

 course on the probability laws for the deviations of the elements and 

 on the functional formula connecting the contemplated transmission 

 characteristic with the elements. 



Before the elements of the system have been manufactured there 

 may, on the other hand, arise the "inverse" problem of setting such 

 restrictions on the manufacturing deviations of the elements as to 

 insure that the contemplated characteristic of the proposed transmis- 

 sion system will have a preassigned probability of lying within a 

 certain specified range. As might be expected, this "inverse" prob- 

 lem is more difficult than the "direct" problem, and often it can be 

 solved only by successive tentative solutions of the corresponding 

 "direct" problem. 



Finally, after the system has been constructed and tested, there may 

 arise the question as to whether its elements have been correctly con- 



