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LGAN
Figure 2.32
Spanning ellipse enclosing the configuration of two head dimensions vari-
ables.
2.9.2 Concentration ellipse
Consider a continuous random variable, say
Y
, with an unspecified distribution having
a mean
µ
and finite variance
σ
2
.Let
Y
∗
be another continuous random variable having
a uniform distribution defined on the interval
(µ
−
σ
√
3,
µ
+
σ
√
3
)
. It then follows
from the expected value and variance of a uniform distribution that E
(
Y
∗
)
=
µ
and
var
(
Y
∗
)
=
σ
2
.ThisledCramer (1946) to suggest the interval
(µ
−
σ
√
3,
µ
+
σ
√
3
)
(2.21)
to be taken as a geometrical description of the concentration of the (unspecified) distri-
bution of
Y
about its known mean
µ
. The interval (2.21) may be called a
concentration
interval
and is not to be confused with a confidence interval for an unknown param-
eter
(µ
−
σ
√
3
2
µ
. We note that if
Y
has a normal
(µ
;
σ
)
distribution then P
<
Y
<
µ
+
σ
√
3
)
=
0
.
9167.
In practice
µ
and
σ
2
are usually unknown, so, if we have a random sample of size
2
n
then, replacing
µ
and
σ
in (2.21) with their sample counterparts, we have the sample
concentration interval
(
x
−
s
√
3,
x
+
s
√
3
).
(2.22)