MA THEM A TICALANAL YSIS OF RA NDOM NOISE 295 



1.1 The Probability of Exactly A' Electrons Arriving at the 

 Anode in Time T 



The fluctuations in the electron stream are supposed to be random. We 

 shall treat this randomness as follows. We count the number of electrons 

 flowing in a long interval of time T measured in seconds. Suppose there 

 are Ki . Repeating this counting process for many intervals all of length 

 T gives a set of numbers Ki , K^ - • • K » where M is the total number of 

 intervals. The average number v, of electrons per second is defined as 



V = Lim — (1.1-1) 



jvi-*oo MT 



where we assume that this limit exists. As M is increased with T being 

 held fixed some of the A's will have the same value. In fact, as M increases 

 the number of A's having any particular value will tend to increase. This 

 of course is based on the assumption that the electron stream is a steady 

 flow upon which random fluctuations are superposed. The probability of 

 getting A' electrons in a given trial is defined as 



,^^. _ . Number of trials giving exactly K electrons ,. . _. 

 p{K) = Lim ^ ,/ (lA-2) 



Of course p{K) also depends upon T. We assume that the random- 

 ness of the electron stream is such than the probability that an electron 

 will arrive at the anode in the interval {t, t + A/) is vM where M is 

 such that vM « 1, and that this probability is independent of what has 

 happened before time / or will happen after time / + A/. 



This assumption is suflacient to determine the expression for p{K) which is 



p{K) = ^^e-'' (1.1-3) 



This is the "law of small probabilities" given by Poisson. One method 

 of derivation sometimes used can be readily illustrated for the case iv = 0. 



T 



Thus, divide the interval, (0, T) into M intervals each of length M = —. 



A/ is taken so small that vA/ is much less than unity. (This is the "small 

 probability" that an electron will arrive in the interval A/). The prob- 

 ability that an electron will not arrive in the first sub-interval is (1 — vA/). 

 The probability that one will not arrive in either the first or the second 

 sub-interval is (1 — I'A/)^. The probability that an electron will not arrive 

 in any of the M intervals is (1 — v^t)". Replacing M by T/A/ and letting 

 A/ — ^ gives 



