380 Oil Random Occurrences in Space and Time 
T—t 
Thus = V 1 (xxxiv). 
ni 
(T — t\ 
There should be a (Hrect proof of this since I ) is clearly the mean number 
of occari'ences in the interval (T — t), or we ought to be able to show that the mean 
number of occurrences is one leas than the mean number of intervals*. 
The regression of n on t is thus seen to be linear, while that oft on n is hyperbolic. 
Again, 
vi^ 1 v?n"~' (n — 1) ! 
dt 
m \ 711 
Accordingly, o-'-„, ( + «/- = (- — ^) + ^-^^ + 1 , 
\ III J in 
It — t 
and a^j = \/ (xxxv). 
V 111 
The arrays are accordingly heterosccdastic. 
Clearly, Nn = X' i'' ^^-^ C^J + 1 ) e^^dt, 
.'0 III' \ 111 
whence integrating and substituting the value for X' we find 
rj. H'-l (Xxxvi). 
1 + e ™ 
m 
To find (Ti, we have 
1 ( Jo - 1) ! j 
\ (n + 1) {n - 1) !| , ((7^ + 1) ! 
2 \in^ III''-' (11 -2)1/ 1 \fii III'' ill J 1 ! 
* Tlie number of intervals is always one less than the number of occurrences, but then there are 
cases of no or one occurrence (giving uo intervals) which affect the mean number of occurrences. 
