Mathematical Analysis of Random Noise

In this section we use the representations of the noise c u r r e n t s given in section 2.8 to derive some statistical properties of l(t). T h e first six sections a r e concerned with the probability distribution of / ( / ) a n d of its zeros and maxima. Sections 3.7 a n d 3.8 are concerned with t h e statistical p r o p erties of the envelope of / ( / ) . F l u c t u a t i o n s of integrals involving I2{t) a r e discussed in section 3.9. T h e p r o b a b i l i t y distribution of a sine wave plus a noise c u r r e n t is given in 3.10 a n d in 3.11 an a l t e r n a t i v e m e t h o d of deriving the results of P a r t I I I is mentioned. Prof. Uhlenbeck has p o i n t e d o u t t h a t much of the m a t e r i a l in this P a r t is closely connected with t h e t h e o r y of Markoff processes. Also S. C h a n d r a s e k h a r hks written a review of a class of physical problems which is related, in a general way, to t h e present subject." 3.1 T H E DISTRIBUTION OF TILE N O I S E CURRENT 2 3