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2.10.1.2 Example
As an example, let us derive a confidence interval for the mean of
N
mea-
surements: the latter quantity is an unbiased estimator of the expecta-
tion value. Assume that the
N
measurements are realizations of a random
Gaussian variable
G
of mean
µ
and standard deviation
σ
. Using the distrib-
utions discussed in the
ne
xt section, it can easily be shown that the random
variable (
M
µ
)
/
(
σ/
√
N
) has a normal distribution, and that the variable
−
n
=1
((
G
1
−
M
)
2
)
/
(
σ
2
) has a Pearson distribution with
N
−
1 degrees of
freedom.
From the definition of the Pearson variable, one might conclude that the
above variable has
N
(not
N
1) degrees of freedom. One should note that
the random variable
M
depends on the random variables
G
i
−
since one has
M
=
n
=1
G
i
/N
: hence the variable has only
N
1 degrees of freedom.
Those variables are independent. From a theorem stated below, the ran-
dom variable
−
N
(
N
µ
i
=1
(
G
i
−
M
−
Z
=
−
1)
M
)
2
has a Student distribution with
N −
1 degrees of freedom. One can easily
compute the value of
z
1
and
z
2
such that a realization of the random variable
Z
lie between those two values with probability 1-
α
,where
α
is a known
quantity (e.g.,
α
= 0.05 if a 95% confidence interval is sought). The quantity
N
(
N
µ
i
=1
(
g
i
−
m
−
z
=
−
1)
,
m
)
2
where
m
is the mean of the
N
measurements
g
i
,andwhere
µ
is the only
unknown, is a realization of the random variable
Z
. Therefore, the only re-
maining task is the resolution of the two inequalities
z
1
<z
and
z<z
2
;they
are linear in
µ
, hence the resolution is trivial. The two solutions
µ
1
=
a
and
µ
2
=
b
are the boundaries of the confidence interval
i
=1
(
g
i
−
i
=1
(
g
i
−
m
)
2
m
)
2
a
=
m
−
z
2
and
b
=
m
−
z
1
.
N
(
N
−
1)
N
(
N
−
1)
Because the Student distribution is symmetrical,
z
1
and
z
2
may be chosen
symmetrically, e.g.,
z
1
=
−
z
0
<
0and
z
2
=
z
0
>
0. The confidence interval is
symmetrical around
m
:
i
=1
(
g
i
−
i
=1
(
g
i
−
m
)
2
m
)
2
a
=
m
−
z
0
and
b
=
m
+
z
0
,
N
(
N
−
1)
N
(
N
−
1)
where
m
,
and
N
depend on the experiments, and
z
0
depends on the
chosen value of
α
only.
{
g
i
}
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