## Expected Value of the Signal for Constant Flux Make the substitution  a " — ST "n

 Equation (3-4) can be written as — ‘ 00 V2(t) ■ E (v t 2 v2(aAt) a = 0 U_ai 00 00 . * E 23 77 L 77 t v(aAt) v(bAt) a = 0 b = 0 Sf ‘ a m ‘ b
 a* b

by separating the terms in which a = b from those in which а ф b, and the expected value of V (t). is

■ 00 ‘

<v2(t)> = <(vt f > y2<aAt>

a = о V Ж ‘ а/

‘ 00 00

+ <7? t 7) t >v(aAt) v(bAt), (3-5)

a = о b = о Ж " a Ж ‘ b а Ф b

since the expectation of the sum of a number of random variables is equal to the sum of their expectations.

If the average flux of neutrons is Ф and the counting efficiency of the detector is k,

then the probability of detecting a neutron during the ntb time interval is кФДі, the probability

th

of detecting no neutron during the n time interval is l-кФді, and the expected values of the Tj combinations are

< h7t 2> = [l2 x k<I>At] + [o2 x (l-k^At)]

Ы — a]

= кФді, <rjt 77 t > = [l2x (кФдО2] + [l x k\$At x 0 x (1-кФдо]

+ [o x (l-k\$At) x 1 x кФді] + [о2 х (1-кФд02] = (кФді) ,

 У ] кФді v2(aAt) + ^ ^ * (кФдІ)2 v(aAt) v(bAt),    so

We have postulated a d-c removing component, such as a series capacitor or a shunt inductor, in the linear part of the circuit and it is easy to show that under this condition

 which becomes, as Д1—*0,     .oo. ‘■ . ‘ • .

v(x) dx = 0. (3-11)

’o.  So the expected value of V^(t) is

(3-13) Also, the output of the integrating circuit, т, satisfies the differential equations   dS(t) + S(t) = AV^(t)

dt t

d<S(t)> + <S(t) > = <AVz(t)> dt t t  and since <AV^(t)> is constant in time, then

If there are two different types of particles present, arguments similar to the above will

yield (3-15)  where the subscripts 1 and 2 refer to the two types of particles. Because of Equation (3-11); this becomes