332:321
Probability and Stochastic Processes
Examination 2
November 22, 2004
You have 80 minutes to answer the following questions in the notebooks provided. You are permitted 1 doublesided
sheet of notes.
Make sure that you have included your name, Rutgers netid and signature in each book used.
(5
points) Read each question carefully. All statements must be justified. Computations should be simplified as much as
possible. The following table may be helpful:
x
0
.
0
0
.
25
0
.
5
0
.
75
1
.
0
1
.
25
1
.
50
1
.
75
2
.
0
2
.
25
2
.
50
(
x
)
0
.
50
0
.
599
0
.
692
0
.
773
0
.
841
0
.
894
0
.
933
0
.
960
0
.
977
0
.
988
0
.
994
1.
40 points
SHORT ANSWER: Each part is a separate problem.
(a)
Y
is a Gaussian
(μ
=
2
, σ
=
2
)
random variable. Calculate
P
[
Y
>
3].
(b)
X
1
,
X
2
and
X
3
are independent identically distributed (iid) continuous uniform random variables. Random
variable
Y
=
X
1
+
X
2
+
X
3
has expected value
E
[
Y
]
=
0 and variance
σ
2
Y
=
9. What is the PDF
f
X
1
(
x
)
of
X
1
?
(c)
X
is a Gaussian
(μ
=
0
, σ
=
1
)
random variable.
Z
is a Gaussian
(
0
,
4
)
random variable.
X
and
Z
are
independent. Let
Y
=
X
+
Z
. Find the correlation coefficient
ρ
of
Z
and
Y
. Are
Z
and
Y
independent?
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 Variance, Probability theory

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