Determination of the Photon Count Rate, λ
Let Q(N, t, λ) dλ = the probability that, given N counts in time interval t, the true count
rate is between λ and λ + dλ. We expect a priori that Q(N, t, λ)  PN ( λ, t), the
probability that a count rate of λ, observed over a time interval t produces N counts.
Hence we can write
Q(N, t, λ) = kPN ( λ, t).
But
e   t ( t ) N
PN ( , t ) 
,
N!
and normalization requires that

1   Q ( N , t ,  )d 
0
k   t
k  x N
k
N
e
(

t
)
d


e x dx  .


0
0
N!
tN!
t
So k = t and therefore
Q( N , t ,  ) 
te  t (t ) N
 tPN ( , t).
N!
From this result we can infer the mean, or expectation value for λ:


0
0
    Q ( N , t ,  )d  
e  t (t ) N 1 d
1

N!
N !t

For large N
  
N 1 N
 .
t
t

0
e  x x N 1dx 
( N  1)! N  1

.
N !t
t