Name:
Winter 2006 Quiz # 5 ECE S-510
You are given a set of N independent identically distributed random variables x1, x2, ….,
xN, from the distribution
η(µ,1)
.
(a) Show that the sufficient statistic associated with the parameter
µ
is T(x1, x2, …., xN)
=
∑
=
N
ii
x
1
and find its probability density function p(T(x1, x2, …., xN)|
µ
).
(b) Find p(x1, x2, …., xN|T(x1, x2, …., xN),
µ
). What do you observe about p(x1, x2, ….,
xN|T(x1, x2, …., xN),
µ
)? How would you go about generating a set of realizations of
x1, x2, …., xN without knowledge of
µ
?
(c) Find a minimum variance unbiased estimator for
µ
and show how you got it.
Fact: If X = ( x1, x2, …., xN) is a N-D vector which is normally distributed with mean
µ
1
and covariance matrix
[Σ] = σ
2[Ι]
, then p(X) =
2
2
)2(
1
N
πσ
exp{-
2
2
1
σ
2
1
)
(
µ
−
∑
=
n
ii
x
}
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