EML 6934
Fall 2009
Optimal Estimation
University of Florida
Mechanical and Aerospace Engineering
HW
4
Issued: September 18, 2009, Due: September 25, 2009 (in class)
Note:
Your work for the questions with “[0 pt]” are not to be turned in.
Problem 1.
[5 pt]
Show that
var
(
aX
) =
a
2
var
(
X
) (
a
is a deterministic parameter).
Problem 2.
[10 + 10 = 20 pt]
Prove the CBS (CauchyBunyakovskiiSchwarz) Inequality for random variables. That is, show
that for two random variables
X
and
Y
,
Covar
(
X, Y
)
≤
radicalbig
V ar
(
X
)
V ar
(
Y
)
and similarly

E[
XY
]

2
≤
E[
X
2
] E[
Y
2
]
Problem 3.
[10 + 10 = 20 pt]
Show the following triangle inequality for random variables, that for two random variables
X
and
Y
,
radicalbig
E[(
X
+
Y
)
2
]
≤
radicalbig
E[
X
2
] +
radicalbig
E[
Y
2
]
and
radicalbig
V ar
(
X
+
Y
)
≤
radicalbig
V ar
(
X
) +
radicalbig
V ar
(
Y
)
Problem 4
(Sample variance as an estimate of the variance)
.
[5+10+0 = 15 pt]
Let
X
1
, . . . , X
n
be
n
independent random variables that have the same mean
μ
= E[
X
i
] and
variance
σ
2
= E[(
X

μ
)
2
], and
μ <
∞
, σ
2
<
∞
. Now consider the
sample mean
ˆ
μ
and the
sample
variance
ˆ
σ
2
:
ˆ
μ
=
X
1
+
X
2
+
. . . X
N
N
,
ˆ
σ
2
=
(
X
1

ˆ
μ
)
2
+ (
X
2

ˆ
μ
)
2
+
· · ·
+ (
X
N

ˆ
μ
)
2
N

1
,
1. What is the variance of the sample mean?
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 Fall '08
 Staff
 Variance, Probability theory, Covariance matrix, xxt

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