STAT 116
STANFORD UNIVERSITY
Department of Statistics
FINAL EXAMINATION
STATISTICS 116
Instructor: Professor Anthony D’Aristotile
August 19, 2006
3:30 pm  6:30 pm
IMPORTANT: Please, show all your work and justify your assertions. There are
200 total points.
STUDENT Name:
STUDENT ID :
1
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1. (20 points)
Let
X
1
, X
2
,
.....
, X
100
be independent Poisson random variables with mean equal to 1. Use
the central limit theorem to approximate
P
(
∑
100
1
X
i
>
75)
2. (20)
From a signpost that says OHIO two letters fall off. A friendly drunk puts the two letters
back into the two empty slots at random.
What is the probability that the sign still says
OHIO?
3. (7 + 7 + 6)
Let
X, X
1
, X
2
, ..., X
n
be identically distributed discrete random variables.
(a) Find
E
(
X

X
=
k
)
.
(b) Find
E
(
X

X
)
(c) Find
E
(
X
1
+
X
2

X
1
+
X
2
+
X
3
+
X
4
+
X
5
)
.
4. (5 + 10 + 10 + 5)
Let
X
and
Y
be independent standard normal random variables.
(a) Find the probability density function (p.d.f.) of
(
X, Y
)
.
(b) Find the p.d.f. of
(
U, V
)
where
U
=
Xcos θ

Y sin θ
V
=
Xsin θ
+
Y cos θ
(c) Find the p.d.f.’s of
U
and
V
.
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 Summer '07
 Staff
 Statistics, Normal Distribution, Probability, Variance, Probability theory, probability density function

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