332:226
Probability and Stochastic Processes
Examination 1
March 12, 2009
SOLUTION – VERSION 1
You have 110 minutes to answer the following
four
questions in the notebooks provided.
This is a closed
book exam; neither notes nor calculators are permitted
. Make sure that you have included your
name, Rutgers netid and signature in each book used (5 points). Leave your exam paper stapled to your
exam blue book (5 points). Read each question carefully. All statements must be justified. Computations
should be simplified as much as possible.
1.
50 points
SHORT ANSWERS: You must explain your answers to these unrelated questions.
(a) A Starburst candy package contains 10 individual candy pieces. Each piece is equally likely to
be red, orange, yellow or pink, independent of all other pieces. What is the probability that a
Starburst package has only red or pink pieces and zero orange or yellow pieces?
Each piece is red or pink with probability
p
= 1
/
2
. The probability of only red or pink pieces is
p
10
.
(b) For events
A
and
B
,
P
[
A
∪
B
] =
P
[
A
] +
P
[
B
]. What is
P
[
A

B
]?
In general,
P
[
A
∪
B
] =
P
[
A
]+
P
[
B
]

P
[
AB
]
. Hence
P
[
AB
] = 0
and thus
P
[
A

B
] =
P
[
AB
]
/P
[
B
] =
0
.
(c)
K
is a geometric (
p
= 1
/
2) random variable. What is
P
[
K
≤
E
[
K
]]?
K
has PMF
P
K
(
k
) =
(
(1

p
)
k

1
p
k
= 1
,
2
, . . .
0
otherwise
=
(
(1
/
2)
k
k
= 1
,
2
, . . .
0
otherwise
Recalling that
E
[
K
] = 1
/p
= 2
, we have
P
[
K
≤
E
[
K
]] =
P
[
K
≤
2] =
P
K
(1) +
P
K
(2) = 1
/
2 + 1
/
4 = 3
/
4
.
(d) The temperature
T
in this thermostaticallycontrolled lecture hall is a Gaussian random variable
with expected value
μ
= 68 degrees Fahrenheit. In addition
P
[
T <
66] =
Q
(1) = 1

Φ(1). What
is the variance of
T
?
P
[
T <
66] =
P
T

68
σ
T
<
66

68
σ
T
= Φ

2
σ
T
=
Q
2
σ
T
=
Q
(1)
.
Thus
σ
T
= 2
and
T
has variance
Var[
T
] =
σ
2
T
= 4
.
(e) MULTIPLE CHOICE: For the course textbook (Yates & Goodman 2nd edition) did you (A) buy
a new copy from a bookstore or online bookseller, (B) buy a used copy, (C) beg, borrow or steal
a copy, (D) survive without a copy, (E) do something else (if so, then what?). Except for answer
(E), you need not give any explanation. All students providing an answer will receive full credit
for this question. By the way, it’s perfectly OK and completely logical to save money by buying,
borrowing or begging (but not stealing) a used copy.
I’m hoping you didn’t answer (A). For each new copy sold, I have to donate my royalty back to
Rutgers accoding to a new NJ state ethics code. I’m looking forward to reading any explanations
for answer (E).
2.
40 points
In a youth basketball league, a player is fouled in the act of shooting a layup. There is a
probability
q
= 0
.
2 that the layup is good, scoring 2 points. If the layup is good, the player is also
awarded 1 free throw, giving the player a chance at a “three point play.” If the layup is missed, then
(because of the foul) the player is still awarded one point automatically and is also awarded one free
throw, enabling a chance to score two points in total. The player makes a free throw with probability
p
= 1
/
2. Let
L
denote the event that the layup is good and
T
denote the event that the free throw is
good. Let
L
c
and
T
c
denote the complementary events.
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 Spring '08
 Staff
 Probability, Probability theory, Lance

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