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If we assume that the elements in the noise vector,
, are independent and normally distributed
^{3.15}random variables with
and
, (which is
not terribly unreasonable to do since noise can come from all kinds of
sources and once we add them all up, the central limit theorem kicks
into effect,) then we can determine if
is
biased and what its variance is.

Before we start, however, we will note that the assumption that
are independent and identically distributed
normal(0, ) variables
implies that **Y** is also
normally distributed with mean
and
variance . This is because
and
functions as
a location parameter.

First, we will show that
is unbiased.

Now we will derive the variance of
.
However, before we get into it, let me first point out that ** E**
. This
can be easily shown using the facts that ** Var**
, ** E**
and the definition of variance. That is,

With that little bit of extra information in hand, we are now ready to
derive the variance of
.

An alternative and shorter derivation of this same variance is as
follows:

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** Up:** Linear Models
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Frank Starmer
2004-05-19