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Assignment 5, STAT220Winter 2011
Instructor: Kamyar Moshksar
Due on
April 4th, In class
Problem 1
 Let
(
X
1
, X
2
, X
3
)
∼
Mult(
n, p
1
, p
2
, p
3
)
.
(a) Compute
Cov(
X
1
+ 3
X
2
, X
2
+
X
3
)
.
(b) Let
α
be a real number. Determine the value of
α
such that
X
1
and
X
2
+
αX
3
are uncorrelated.
(c) Calculate
ρ
X
1
,X
2
2
.
(d) Write down the formulae for
f
X
1
,X
2

X
1
+
X
3
(
., .

t
)
where
t
∈
R
X
1
+
X
3
.
Problem 2
Let
X
be a random variable whose moment generating function is
M
X
(
t
) =
a
2

e

t
.
where
a
∈
.
(a) Find
a
.
(b) Determine values of
t
∈
such that the definition for
M
X
(
t
)
is meaningful.
(c) Find
E
(
X
(
X
2
+ 1))
.
(d) Find
f
X
(
.
)
.
(e) Find
P
(
X
is an even number
)
.
Problem 3
 In this problem we are going to see why the sum of two independent random variables
X
1
∼ N
(
μ
1
, σ
2
1
)
and
X
2
∼ N
(
μ
2
, σ
2
2
)
is again a normal random variable.
(a) Recall that for two independent discrete random variables
W
1
and
W
2
we showed in class that
E
[
g
(
W
1
)
h
(
W
2
)] =
E
[
g
(
W
1
)]
E
[
h
(
W
2
)]
holds for “any” two functions
g
(
.
)
and
h
(
.
)
. In fact, the same statement holds for any two independent
continuous random variables
Z
1
and
Z
2
. Use this fact to show that
M
Z
1
+
Z
2
(
t
) =
M
Z
1
(
t
)
M
Z
2
(
t
)
.
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(b) Use the statement proved in part (a) to show that
X
1
+
X
2
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 Fall '08
 Burbidge,John
 Normal Distribution, Probability theory, probability density function, Kamyar Moshksar

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