NAME:
ECE 6540 MIDTERM 1
Show your work. Closed book, limited notes (1 page). No laptops or calculators.
1. (
25 points
) We have
N
observations
x
[
n
] =
w
[
n
],
n
= 0
,
1
,...N

1 where
w
[
n
] is IID
(independent, identically distributed) with an uniform distribution
U
[0
,β
].
(a) Show that
ˆ
β
=
2
N
∑
N

1
n
=0
x
[
n
] is an unbiased estimator for
β
.
(b) Let
σ
2
denote the unknown variance for our uniform distribution. It is known
that the variance of a uniform distribution of the form
U
[0
,β
] is equal to
β
2
/
12.
Therefore, we could estimate
σ
2
as a function of the estimator from the previous
part:
ˆ
σ
2
=
ˆ
β
2
12
Is this estimator for
σ
2
unbiased? What happens as
N
→ ∞
?
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2. (
25 points
) A credit card company’s fraud prevention software monitors all transactions
and generates an alert whenever suspicious activity is detected. Let
x
[
n
] denote the
number of alerts generated on day number
n
. If we record the number of alerts for
N
days,
x
[
n
] for
n
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 Brown
 Normal Distribution, Probability theory, Estimation theory, Fisher Information, CramerRao lower bound

Click to edit the document details