1
ISyE 6739 — Test #2 Solutions
Summer 2005
This test is open notes, open books. You have
exactly 90 minutes
. Just do the best you
can, and good luck!
1. (30 points) Short Answer Questions.
(a) Suppose
X
has p.d.f.
f
(
x
) = 5
x
4
, 0
≤
x
≤
1. Find
E
[2
X

5].
ANSWER: 2
E
[
X
]

5 =

10
/
3.
(b) If
X
again has p.d.f.
f
(
x
) = 5
x
4
, 0
≤
x
≤
1, ﬁnd
Var
(2
X

5).
ANSWER: 4
Var
(
X
) = 0
.
0793.
(c) Suppose
X
can equal 1 or 2, each with probability 1
/
2. Find
E
[
‘
n(
X
)].
ANSWER:
1
2
‘
n(2) = 0
.
347.
(d) TRUE or FALSE?
E
[
X
2
]
≥
(
E
[
X
])
2
.
ANSWER: True.
(e) Suppose
X
has m.g.f.
M
X
(
t
) = 0
.
3
e
t
+ 0
.
7. What’s the distribution of
X
?
ANSWER: Bern(0.3).
(f) If
X
has m.g.f. 4
/
(4

t
), for
t <
4, ﬁnd the m.g.f. of 2
X

1.
ANSWER: 4
e

t
/
(4

2
t
),
t <
2.
(g) TRUE or FALSE? If
Cov
(
X,Y
) = 0, then
X
and
Y
are independent.
ANSWER: False.
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(h) Suppose that
X
and
Y
are independent Exponential(
λ
) random variables.
What is the m.g.f. of
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 Spring '10
 tricahuu
 Probability theory, var, short answer questions., Joey Yi

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