A simple, but interesting, example that results in a
computable answer occurs when the mean of Gaussian random
variables is either zero (model 0) or is
±m
±
m
with equal probability (hypothesis 1). The second hypothesis
means that a nonzero mean is present, but its sign is not
known. We are therefore stating that if hypothesis one is in
fact valid, the mean has fixed sign for each observation 
what is random is its a priori value. As
before,
L
L
statistically independent observations are made.
ℳ
0
:
r∼𝒩0σ2I
ℳ
0
:
r
0
σ
2
I
ℳ
1
:
r∼𝒩mσ2I
ℳ
1
:
r
m
σ
2
I
m=m…mProb1/2−m…−mProb1/2
m
m
…
m
12
m
…
m
12
The numerator of the likelihood ratio is the sum of two
Gaussian densities weighted by
1/2 12 (the
a priori probability values), one having a
positive mean, the other negative. The likelihood ratio,
after simple cancellation of common terms, becomes
Λr=12e2m∑l=0L−1
r
l
−Lm22σ2+12e2m∑l=0L−1
r
l
−Lm22σ2
Λ
r
1
2
2
m
l
0
L
1
r
l
L
m
2
2
σ
2
1
2
2
m
l
0
L
1
r
l
L
m
2
2
σ
2
and the decision rule takes the form
coshmσ2∑l=0L−1
r
l
≷
ℳ
0
ℳ
1
(ηeLm22σ2)
m
σ
2
l
0
L
1
r
l
≷
ℳ
0
ℳ
1
η
L
m
2
2
σ
2
where
coshx
x
is the hyperbolic cosine given simply as
ex+e−x2
x
x
2
. As this quantity is an even function, the sign of
its argument has no effect on the result. The decision rule
can be written more simply as
∑l=0L−1
r
l

≷
ℳ
0
ℳ
1
(σ2marccoshηeLm22σ2)
l
0
L
1
r
l
≷
ℳ
0
ℳ
1
σ
2
m
η
L
m
2
2
σ
2
The sufficient statistic equals the
magnitude of the sum of the observations
in this case. While the right side of this expression, which
equals
γ γ, is complicated, it need only
be computed once. Calculation of the performance
probabilities can be complicated; in this case, the
falsealarm probability is easy to find and the others more
difficult.