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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 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
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
σ 3
2
µ
. We note that if Y
has a normal
; 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)
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