MATHEMATICAL THEORY OF COMMUNICATION 625 



4. Let points be distributed on the / axis according to a Poisson distribu- 

 tion. At each selected point the function /(/) is placed and the different 

 functions added, giving the ensemble 



■11 fit + h) 



where the tk are the points of the Poisson distribution. This ensemble 

 can be considered as a type of impulse or shot noise where all the impulses 

 are identical. 



5. The set of English speech functions with the probability measure given 

 by the frequency of occurrence in ordinary use. 



An ensemble of functions /„(/) is stationary if the same ensemble results 

 when all functions are shifted any fixed amount in time. The ensemble 



fe{t) = sin {t + e) 



is stationary if B distributed uniformly from to 2ir. If we shift each func- 

 tion by /i we obtain 



U{t + /i) = sin (/ + /i -h 0) 



= sin (/ + (p) 



with ip distributed uniformly from to 2ir. Each function has changed 

 but the ensemble as a whole is invariant under the translation. The other 

 examples given above are also stationary. 



An ensemble is ergodic if it is stationary, and there is no subset of the func- 

 tions in the set with a probability different from and 1 which is stationary. 

 The ensemble 



sin (/ + 0) 



is ergodic. No subset of these functions of probability t^O, 1 is transformed 

 into itself under all time translations. On the other hand the ensemble 



a sin (/ -|- 6) 



with a distributed normally and 6 uniform is stationary but not ergodic. 

 The subset of these functions with a between and 1 for example is 

 stationary. 



Of the examples given, 3 and 4 are ergodic, and 5 may perhaps be con- 

 sidered so. If an ensemble is ergodic we may say roughly that each func- 

 tion in the set is typical of the ensemble. More precisely it is known that 

 with an ergodic ensemble an average of any statistic over the ensemble is 

 equal (with probability 1) to an average over all the time translations of a 



