Entropy 
.
These results appear in a different form in equations 9.36
and 9.37.
. If 
, 
,
then
16.1
16.2
Mutual information 
.
Define 
.
. If the 
are non-negative integers (the integer condition is used only in the simplification
of the term in theorem 17.2; for the other terms it may be relaxed) then
17.1
where
17.2
where
To find 
substitute 
for 
and 
for 
in the expression for 
,
letting 
and 
and let 
.
To find 
substitute
for ,
let ,
and note that the range on the changed summation affected changes from
m to n, and substitute
for ,
all of this in the expression for 
.
where 
is defined in theorem 15a.
Proof: (17.1) Write the mutual information as the sum of three entropies 
, 
,
and 
and apply theorem 13.1. (17.2) Square the mutual information written as
the sum of three entropies and make the obvious identifications necessary
for each term. Apply theorems 13.1, 13.2, 13.3, 15a.3, and 15b.3 as needed.
QED.
Average 
.
. If , ,
then
18.1
18.2
Variance 
.
Note that 
.
where 
; 
is not the variance of the estimator 
.
. If , ,
then
19.1
19.2
Proof: The integrals may be found by applying theorem 12.
Covariance 
.
. If , ,
then
Proof: Apply theorem 14a.
The second posterior mean, the estimator for the second moment, of the covariance depends on multiple overlap integrals and is given in reference [95], theorem 26.
Chi-squared 
.
. If 
, 
, 
, , 
then
Proof: Apply theorem 14a.