Statistics 400 / Mathematics 463
Practice Exam 1 (BL1) Spring 2012
75 min
Version A
Solutions
1. True or False (No need to explain)
[14 points]
(a) For a continuous random variable, the density function should be continuous.
F. For a continuous random variable, cdf should be continuous, not pdf.
(b) Standard Cauchy distribution has a density
f
(
x
) =
1
π
(1 +
x
2
)
that is symmetric
about the origin, so the mean and median both equal zero.
F. Median is 0, but mean does not exist.
(c) Consider two baseball players A and B. It is possible for batter A to have a
higher average than batter B for each season during their careers and yet B
could have a better overall average at the end of their careers.
T. Simpson’s paradox.
(d) Let
X
be of Uniform(a,b) distribution, then the value
c
that minimized
E
(
X

c
)
2
is
c
= (
a
+
b
)
/
2.
T. Mean
EX
is the value to minimize the function
E
(
X

c
)
2
.
(e) If two events A and B are mutually exclusive, then they are also independent.
F. Mutually exclusive implies that
P
(
A
∩
B
) = 0, while independence requires
that
P
(
A
∩
B
) =
P
(
A
)
P
(
B
) which is not necessarily zero.
(f) To scramble the word “ILLINI”, there are 60 different arrangements.
T. Total number of arrangements is
6
3
,
2
,
1
= 60.
(g) Let A and B be two events of positive probability. Then the conditional proba
bility
P
(
A

B
)
> P
(
A
) if we know that
P
(
B

A
)
> P
(
B
).
T.
P
(
B

A
)
> P
(
B
) implied that
P
(
A
∩
B
)
> P
(
A
)
P
(
B
), which in turn implies
that
P
(
A

B
)
> P
(
A
). Actually
P
(
A

B
)
> P
(
A
) if and only if
P
(
B

A
)
> P
(
B
).
1
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2.
[10 points]
Suppose that E and F are two events such that
P
(
E
) = 0
.
3 ,
P
(
F
) = 0
.
5
and
P
(
F

E
) = 1
/
3.
(a)
[5 points]
Find
P
(
F
0

E
0
).
P
(
F

E
) =
P
(
E
∩
F
)
/P
(
E
) = 1
/
3, so
P
(
E
∩
F
) = 0
.
1,
P
(
E
∪
F
) =
P
(
E
) +
P
(
F
)

P
(
E
∩
F
) = 0
.
7 and
P
(
E
0
∩
F
0
) =
P
((
E
∪
F
)
0
) = 1

0
.
7 = 0
.
3.
Therefore
P
(
F
0

E
0
) =
P
(
E
0
∩
F
0
)
/P
(
E
0
) = 0
.
3
/
0
.
7 = 3
/
7.
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 Spring '08
 Kim
 Statistics, Normal Distribution, Probability, Probability theory

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