EML 6934
Fall 2009
Optimal Estimation
University of Florida
Mechanical and Aerospace Engineering
HW
3
Issued: September 11, 2009, Due: September 18, 2009 (in class)
Problem 1.
[5 pt]
If
X
∼
N
(
μ, σ
), compute
P
(

X
−
μ

>
3
σ
). If you need tabulated values of
erf
(
x
), you can find
them in the reading material available in the elearning website (chapter 2)
Problem 2.
[5+5 = 10 pt]
Show that if
Y
∼
N
(
μ, σ
), then E[
Y
] =
μ
and
var
(
Y
) =
σ
2
.
Problem 3.
[10 pt] If
X
is a random variable that is uniformly distributed between
a
and
b
, and
γ
is a real constant, show that
γX
is uniformly distributed between
γa
and
γb
. (hint: define
Z
Δ
=
γX
.
Then
F
Z
(
z
) =
P
(
Z
≤
z
) =
P
(
X
≤
z/γ
) =
F
X
(
. . .
) =
..
)
Problem 4.
[10 pt]
Compute the mean and variance of a random variable
X
that is uniformly distributed between
a
and
b
.
Problem 5.
[5 pt]
Show that for a r.v.
X
,
V ar
(
X
) = E[
X
2
]
−
(
¯
X
)
2
.
[5 pt]
Problem 6.
[5+5 = 10 pt]
The unit square
U
in
R
2
is the region such that the points in this region have their
x
 and
y

coordinates between 0 and 1. Let
X
be random vector that is uniformly distributed in the unit square.
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 Fall '08
 Staff
 Probability theory, UK

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