Statistics 116  Fall 2004
Theory of Probability
Midterm # 2, Practice # 1
show (and briefly explain) all of
your work.
calculators are permitted for
numerical calculations only.
Instructions:
Answer 4 out of 5 questions. Clearly mark which 4 questions
you decide to answer. If you do not clearly indicate which 4 are to be counted,
your mark will be based on 5 instead of 4 questions, there are no bonus points.
All questions have equal weight.
Q. 1) Let
X
be a Binomial random variable with parameters
n
and
p
.
Show
that
E
1
X
+ 1
=
1

(1

p
)
n
+1
(
n
+ 1)
p
.
Solution:
E
1
X
+ 1
=
n
j
=0
1
j
+ 1
n
j
p
j
(1

p
)
n

j
=
n
j
=0
n
!
(
n

j
)!(
j
+ 1)!
p
j
(1

p
)
n

j
=
1
(
n
+ 1)
p
n
j
=0
(
n
+ 1)!
(
n

j
)!(
j
+ 1)!
p
j
+1
(1

p
)
n

j
=
1
(
n
+ 1)
p
n
+1
j
=1
(
n
+ 1)!
(
n
+ 1

j
)!(
j
)!
p
j
(1

p
)
n
+1

j
=
1
(
n
+ 1)
p
(
1

(1

p
)
n
+1
)
1
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Q. 2) A filling station is supplied with gasoline once a week. Suppose its weekly
volume of sales in thousands of gallons is a random variable with proba
bility density function
f
(
x
) =
5(1

x
)
4
0
< x <
1
0
otherwise.
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 Spring '11
 Teerana
 Probability theory, probability density function, Cumulative distribution function, dy dx, dx

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