MATHEMATICAL ANALYSIS OF RANDOM NOISE 107 



which is a special case of 



ab(y-\- 1)(1 -z?,Fi(a+ l,i+ l;c;s) = AoF^ia, b; c;z) 



- (7- i)(c-a)(c- b)2Fi(a- l,b- \;c;z) (3.10-30) 



where y = c — a — b and 



^1 = (7' - Ih + (1 - 3)[(7 - l)(c - b)(b - 1) + (7 + \)a(c - a - 1)] 



Although this expression does not show it, A is really symmetrical in a 

 and b. A symmetrical form may be obtained by using the expression ob- 

 tained by putting s = in (3.10-30). 



3.11 Shot Effect Representation 



In most of the work in this part the representations (2.8-1) or (2.8-6) 

 have been used as a starting point. Here we point out that the shot effect 

 representation used in Part I may also be used as a starting point. 



For example, suppose we wish to find the two dimensional distribution of 

 /(/) and /(/ + t) discussed in Section 3.2. This is a special case of the distri- 

 bution of the two variables 



(3.11-1) 



F{t) dt = / G{t) dt = (3.11-2) 



in order that the average values of I and / may be zero. In fact, to get 

 /(/ + r) from /(/) we set G{t) equal to F(t -f r). 



The distribution of / and / may be obtained in much the same manner 

 as was the distribution of / alone in section 1.4. The characteristic func- 

 tion of the distribution is 



f{u, v) = ave. e'"'^'"' 



(3.11-3) 



= exp V f [e»-»^^')+-«(') _ 1] di 

 J— 00 



where v is the expected number of events (electron arrivals in the shot effect) 

 per second. The probability density of / and / is 



~f du f dve~''''-''"f(u,v) (3AI-4) 



Air- J-co J-x 



