UCLA Econ 41 (Spring 2015) Department of Economics, Middle Term Exam (Version A)
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Econ 41 (Spring 2015)
Department of Economics, UCLA
Middle Term Exam (Version A)
The problems in this section only have one correct answer among the choices a, b, c and d. For
each problem, you will get 1 credit if your choice is correct and 0 credit otherwise. Please write
down the version (i.e., A or B) of your exam paper in your scantron. If the version of
your exam paper is missing, your scantron will be graded three times (by assuming it was A and
B respectively) and your grade of the exam will be the smallest one among the three.
The following formula may be useful:
(a + a
2 + a
3 + a
4 + ) = X1
k=1
a
k =
a
1
a
; for any jaj < 1;
(1 + a + a
2 + a
3 + ) = X1
k=0
a
k =
1
1
a
; for any jaj < 1;
(a + 2a
2 + 3a
3 + 4a
4 + ) = X1
k=1
kak =
a
(1
a)
2
; for any jaj < 1;
(a
2 + 2a
3 + 3a
4 + 4a
5 + ) = X1
k=1
kak+1 =
a
2
(1
a)
2
; for any jaj < 1:
You are supposed write down your name and UID on both the Scantron and the exam
paper. The exam paper must be turn in together with the Scantron for the full consideration
of your grade.
Name:_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
UID:_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Multiple Choice Problems
The problems in this section only has one correct answer among the choices a, b, c and d. You will
get 1 credit if your choice is correct and 0 credit otherwise.
A. Roll a 4sided fair die and observe the number on the upper face. The outcome space is
O = f1; 2; 3; 4g and each outcome has the same probability. Suppose we roll the die 3 times
independently and record the number we observe in each trial. Let X denote the number of
times we observe 1s.
(1) Whatís the probability of X 2?
(a) 0.9844 (b) 0.9492 (c) 0.7383 (d) 0.8438
(2) Whatís the probability of X > 1?
(a) 0.1156 (b) 0.0508 (c) 0.2617 (d) 0.1562
(3) DeÖne a new random variable Y = X2
. Whatís the mean of Y ?
(a) 12
16 (b) 32
16 (c) 18
16 (d) 9
16
(4) DeÖne a new random variable Y = (X
2)2
. Whatís the mean of Y ?
(a) 9
16 (b) 44
16 (c) 12
16 (d) 34
16
B. Flip a fair coin sequentially and independently until one observes heads. Let X denote the
number of tails one observes before he/she stops.
(5) Whatís the probability of X = 0?
(a) 0.0 (b) 0.5 (c) 0.6 (d) 0.8
(6) Whatís the probability of X > 4?
(a) 15
16 (b) 1
32 (c) 31
32 (d) 1
16
(7) Whatís the mean of X?
(a) 1 (b) 2 (c) 3 (d) 4
(8) Suppose we know that X 4. Whatís the probability of X 5?
(a) 1.00 (b) 0.25 (c) 0.75 (d) 0.50
C. We draw 4 cards from a deck of 52 playing cards.
(9) What is the probability of getting 4 cards with the same shape?
(a) 0.0106 (b) 0.1518 (c) 0.0026 (d) 0.1122
(10) What is the probability of getting 4 cards with two black cards and two red cards?
(a) 0.0810 (b) 0.3902 (c) 0.1225 (d) 0.5000
(11) What is the probability of getting 4 cards with at least one King?
(a) 0.0769 (b) 0.9231 (c) 0.7187 (d) 0.2813
D. Suppose that A and B are two events.
(12) If A and B are mutually exclusive, and P(A) = 0:4, what is P(B0
jA)?
(a) 1.00 (b) 0.25 (c) 0.75 (d) 0.50
(13) If A and B are mutually exclusive, P(A) = 0:4 and P(A0 \ B0
) = 0:5, what is P(B)?
(a) 0.10 (b) 0.60 (c) 0.40 (d) 0.50
(14) If A and B are mutually exclusive, P(A) = 0:4 and P(B) = 0:1, what is P(A [ B)?
(a) 0.20 (b) 0.30 (c) 0.50 (d) 0.70
(15) If A and B are exhaustive and P(A \ B) = 0:5, what is P(A0 \ B) + P(A \ B0
)?
(a) 0.20 (b) 0.30 (c) 0.50 (d) 0.70
(16) If A and B are independent, P(A) = 0:5 and P(B) = 0:5, what is P(A0 [ B)?
(a) 0.25 (b) 0.45 (c) 0.50 (d) 0.75
E. A and B are two events. We know that P(BjA) = 2
3
and P(AjB) = 1
2
.
(17) If we know that P(A) = 1
2
, what is P(B)?
(a) 1
3
(b) 1
4
(c) 2
3
(d) 1
6
(18) If we know that P(B) = 1
2
, what is P(A)?
(a) 3
5
(b) 1
4
(c) 5
6
(d) 3
8
F. Suppose that there are 5 dollar bills in a box: three 1 dollar bills, one 5 dollar bill and one
10 dollar bill. Two individuals, say T and J, are allowed to pick up one bill from the box
randomly and sequentially (T picks up the bill Örst). Let X denote the money T gets and Y
denote the money J gets.
(19) Whatís the mean of X?
(a) 1.0 (b) 2.5 (c) 3.6 (d) 4.8
(20) Whatís the variance of X?
(a) 12.64 (b) 25.60 (c) 12.96 (d) 23.04
(21) Suppose that the bill that T picks up will not be put back to the box. Whatís the probability
that Y > 1?
(a) 1.0 (b) 0.3 (c) 0.4 (d) 0.8
(22) Suppose that the bill that T picks up will be put back to the box. Whatís the mean of Y ?
(a) 1.0 (b) 2.5 (c) 3.6 (d) 4.8
(23) Suppose that the bill that T picks up will be put back to the box. Whatís the covariance
between X and Y ?
(a) 0.6250 (b) 0.1325 (c) 0.5000 (d) 0.0000
G. Let the random variable X have the p.m.f.
f(x0) = (jx0j + 1)2
9
; x0 = 1;
0; 1:
(24) Whatís the mean of X?
(a) 0.0 (b) 1.5 (c) 1.0 (d) 2.0
(25) Whatís the expectation of X2
2X + 2?
(a) 11
9
(b) 26
9
(c) 15
9
(d) 13
9
H. Suppose that X is a random variable with support S = f2;
1;
0; 1; 2g and we know that
P(X = x0) = x
2
0
12
, for x0 2 f2;
1;
1; 2g:
(26) Whatís the probability that X = 0?
(a) 0 (b) 1
12 (c) 2
12 (d) 2
10
(27) Whatís the probability that of
3
2 X
3
2
?
(a) 1
5
(b) 1
4
(c) 1
2
(d) 1
3
(28) Let Y1 = X + 1. Whatís the mean of Y1?
(a) 1.0 (b) 1.5 (c) 2.5 (d) 3.0
(29) Let Y2 = jXj + 1. Whatís the mean of Y2?
(a) 1.0 (b) 1.5 (c) 2.5 (d) 3.0
(30) If Y1 = X + 1 and Y2 = jXj + 1, whatís the covariance between Y1 and Y2?
(a) 0.0 (b) 0.5 (c) 2.5 (d) 2.0
Department of Economics, UCLA
Middle Term Exam (Version A)
The problems in this section only have one correct answer among the choices a, b, c and d. For
each problem, you will get 1 credit if your choice is correct and 0 credit otherwise. Please write
down the version (i.e., A or B) of your exam paper in your scantron. If the version of
your exam paper is missing, your scantron will be graded three times (by assuming it was A and
B respectively) and your grade of the exam will be the smallest one among the three.
The following formula may be useful:
(a + a
2 + a
3 + a
4 + ) = X1
k=1
a
k =
a
1
a
; for any jaj < 1;
(1 + a + a
2 + a
3 + ) = X1
k=0
a
k =
1
1
a
; for any jaj < 1;
(a + 2a
2 + 3a
3 + 4a
4 + ) = X1
k=1
kak =
a
(1
a)
2
; for any jaj < 1;
(a
2 + 2a
3 + 3a
4 + 4a
5 + ) = X1
k=1
kak+1 =
a
2
(1
a)
2
; for any jaj < 1:
You are supposed write down your name and UID on both the Scantron and the exam
paper. The exam paper must be turn in together with the Scantron for the full consideration
of your grade.
Name:_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
UID:_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Multiple Choice Problems
The problems in this section only has one correct answer among the choices a, b, c and d. You will
get 1 credit if your choice is correct and 0 credit otherwise.
A. Roll a 4sided fair die and observe the number on the upper face. The outcome space is
O = f1; 2; 3; 4g and each outcome has the same probability. Suppose we roll the die 3 times
independently and record the number we observe in each trial. Let X denote the number of
times we observe 1s.
(1) Whatís the probability of X 2?
(a) 0.9844 (b) 0.9492 (c) 0.7383 (d) 0.8438
(2) Whatís the probability of X > 1?
(a) 0.1156 (b) 0.0508 (c) 0.2617 (d) 0.1562
(3) DeÖne a new random variable Y = X2
. Whatís the mean of Y ?
(a) 12
16 (b) 32
16 (c) 18
16 (d) 9
16
(4) DeÖne a new random variable Y = (X
2)2
. Whatís the mean of Y ?
(a) 9
16 (b) 44
16 (c) 12
16 (d) 34
16
B. Flip a fair coin sequentially and independently until one observes heads. Let X denote the
number of tails one observes before he/she stops.
(5) Whatís the probability of X = 0?
(a) 0.0 (b) 0.5 (c) 0.6 (d) 0.8
(6) Whatís the probability of X > 4?
(a) 15
16 (b) 1
32 (c) 31
32 (d) 1
16
(7) Whatís the mean of X?
(a) 1 (b) 2 (c) 3 (d) 4
(8) Suppose we know that X 4. Whatís the probability of X 5?
(a) 1.00 (b) 0.25 (c) 0.75 (d) 0.50
C. We draw 4 cards from a deck of 52 playing cards.
(9) What is the probability of getting 4 cards with the same shape?
(a) 0.0106 (b) 0.1518 (c) 0.0026 (d) 0.1122
(10) What is the probability of getting 4 cards with two black cards and two red cards?
(a) 0.0810 (b) 0.3902 (c) 0.1225 (d) 0.5000
(11) What is the probability of getting 4 cards with at least one King?
(a) 0.0769 (b) 0.9231 (c) 0.7187 (d) 0.2813
D. Suppose that A and B are two events.
(12) If A and B are mutually exclusive, and P(A) = 0:4, what is P(B0
jA)?
(a) 1.00 (b) 0.25 (c) 0.75 (d) 0.50
(13) If A and B are mutually exclusive, P(A) = 0:4 and P(A0 \ B0
) = 0:5, what is P(B)?
(a) 0.10 (b) 0.60 (c) 0.40 (d) 0.50
(14) If A and B are mutually exclusive, P(A) = 0:4 and P(B) = 0:1, what is P(A [ B)?
(a) 0.20 (b) 0.30 (c) 0.50 (d) 0.70
(15) If A and B are exhaustive and P(A \ B) = 0:5, what is P(A0 \ B) + P(A \ B0
)?
(a) 0.20 (b) 0.30 (c) 0.50 (d) 0.70
(16) If A and B are independent, P(A) = 0:5 and P(B) = 0:5, what is P(A0 [ B)?
(a) 0.25 (b) 0.45 (c) 0.50 (d) 0.75
E. A and B are two events. We know that P(BjA) = 2
3
and P(AjB) = 1
2
.
(17) If we know that P(A) = 1
2
, what is P(B)?
(a) 1
3
(b) 1
4
(c) 2
3
(d) 1
6
(18) If we know that P(B) = 1
2
, what is P(A)?
(a) 3
5
(b) 1
4
(c) 5
6
(d) 3
8
F. Suppose that there are 5 dollar bills in a box: three 1 dollar bills, one 5 dollar bill and one
10 dollar bill. Two individuals, say T and J, are allowed to pick up one bill from the box
randomly and sequentially (T picks up the bill Örst). Let X denote the money T gets and Y
denote the money J gets.
(19) Whatís the mean of X?
(a) 1.0 (b) 2.5 (c) 3.6 (d) 4.8
(20) Whatís the variance of X?
(a) 12.64 (b) 25.60 (c) 12.96 (d) 23.04
(21) Suppose that the bill that T picks up will not be put back to the box. Whatís the probability
that Y > 1?
(a) 1.0 (b) 0.3 (c) 0.4 (d) 0.8
(22) Suppose that the bill that T picks up will be put back to the box. Whatís the mean of Y ?
(a) 1.0 (b) 2.5 (c) 3.6 (d) 4.8
(23) Suppose that the bill that T picks up will be put back to the box. Whatís the covariance
between X and Y ?
(a) 0.6250 (b) 0.1325 (c) 0.5000 (d) 0.0000
G. Let the random variable X have the p.m.f.
f(x0) = (jx0j + 1)2
9
; x0 = 1;
0; 1:
(24) Whatís the mean of X?
(a) 0.0 (b) 1.5 (c) 1.0 (d) 2.0
(25) Whatís the expectation of X2
2X + 2?
(a) 11
9
(b) 26
9
(c) 15
9
(d) 13
9
H. Suppose that X is a random variable with support S = f2;
1;
0; 1; 2g and we know that
P(X = x0) = x
2
0
12
, for x0 2 f2;
1;
1; 2g:
(26) Whatís the probability that X = 0?
(a) 0 (b) 1
12 (c) 2
12 (d) 2
10
(27) Whatís the probability that of
3
2 X
3
2
?
(a) 1
5
(b) 1
4
(c) 1
2
(d) 1
3
(28) Let Y1 = X + 1. Whatís the mean of Y1?
(a) 1.0 (b) 1.5 (c) 2.5 (d) 3.0
(29) Let Y2 = jXj + 1. Whatís the mean of Y2?
(a) 1.0 (b) 1.5 (c) 2.5 (d) 3.0
(30) If Y1 = X + 1 and Y2 = jXj + 1, whatís the covariance between Y1 and Y2?
(a) 0.0 (b) 0.5 (c) 2.5 (d) 2.0

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UCLA Econ 41 (Spring 2015) Department of Economics, Middle Term Exam (Version A)