ECON 221 Introduction to Probability and
Statistics I
Midterm II
Solutions
Mirza Troki
c
[email protected]
Department of Economics, Bilkent University, Ankara, Turkey
Suggested solutions to the second midterm.
November 25, 2013
Common Distributions
CD
Econ 221 - Assignment 2 Answer Key
October 3, 2013
1S = fF F; F M; M F; M M g
aA = fF F g
bB = fM M g
cC = fM F; F M; M M g
dA\B =?
eA [ B = fF F; M M g
fA\C =?
gA [ C = fF F; M F; F M; M M g = S
hB \ C = fM M g = B
iB [ C = fM F; F M; M M g = C
jC \ B =
EXAMPLE 1
Organizing & Summarizing your data
An occupational analyst obtained the time (in seconds) for a random sample of n=100 support
staff to complete a particular task. The goal is to complete this task in less than 4 minutes (240
seconds). Results a
CLASS EXAMPLES
CHP 2 Describing Data: Numerical
Ex 1 :
A random sample of 5 business economists was asked to predict the percentage growth in the
consumer price index (CPI) over the next year. The forecast are;
3.6
3.1
a)
b)
c)
d)
3.9
3.7
3.5
Find the mea
CLASS EXAMPLES
CHP 7 & 8 Estimation
Ex 1 :
A polling company is under contract with an advertising agency to update estimates as to the
mean number of hours spent by mid-westerners watching television per week. A sample of 49
persons living in mid-west wh
CLASS EXAMPLES
CHP 4 Discrete Random Variables & Probability Distributions
Ex 1 :
Consider the following probability distribution function (pdf). Let X denotes number of cars
sold in a day.
X
0
1
2
3
4
5
6
P(X)
0.07
0.19
0.23
0.17
0.16
0.14
0.04
a)
b)
c)
Box and Whisker Plot is a graph that describes the shape of the distribution in terms of the five-number summary: the
minimum value, first quartile, median, third quartile, and the maximum value. The inner box shows the numbers that span the
range from th
Econ 221 - Assignment 2 Answer Key
October 3, 2013
1S = fF F; F M; M F; M M g
aA = fF F g
bB = fM M g
cC = fM F; F M; M M g
dA\B =?
eA [ B = fF F; M M g
fA\C =?
gA [ C = fF F; M F; F M; M M g = S
hB \ C = fM M g = B
iB [ C = fM F; F M; M M g = C
jC \ B =
Assingment 3 Solutions
March 24, 2014
1
1. (a) (Dichotomized Regression) First consider E[i ]. In order to compute this expectation
x
we need to know the density of xi f (i ). But since xi is binary random variable it
x
takes only two values:
P (i = 1) =
ECON 221 Introduction to Probability and
Statistics I
Midterm II
Solutions
Mirza Troki
c
[email protected]
Department of Economics, Bilkent University, Ankara, Turkey
Suggested solutions to the second midterm.
November 25, 2013
Common Distributions
CD
Assignment 2 Solutions
April 5, 2014
1
1. Assume p(z) is a random variable. Then Fp (x) = P (p(z) < x). Bu this can be written
as:
P (2(1 (|z|) < x) = P (|z|) > 1
x
2
(1)
= P 1 (|z|) > 1 1
= P |z| > 1 1
x
2
= 1 P |z| 1 1
x
2
(3)
x
2
x
x
z 1 1
2
2
x
Assignment I Solutions
March 9, 2014
1. We have P (A|B) = P (A B)/P (B). But since A B we know that A B = A. Thus P (A B) = P (A)
and nally we get P (A) = P (A)/P (B)
2. Consider the expectation E (Xh(Y )|Y )
This can be wrtitten as E (Xh(Y )|Y ) =
xh(y)f
Assignment 2 Solutions
March 24, 2014
1
1. x E n the norm of
x
x
is given by
1 2
x
x
x
x
x=
2
2
= 1.
Consider two vectors x, y E n . Now we will nd norm of sum of normalized x and y:
y
x
+
x
y
x
y
+
x
y
xx
yy
2x y
+
+
2
x
x y
y 2
xy
= 2 1+
x y
=
(1)
(2)
L
Econ 221 - Assignment 3 Answer Key
March 16, 2014
1)
Ri =the plane is in region i where i=1,2,3
P (Ri ) = 1=3
U=search of region 1 was unsuccessful
P (U nR1 ) = 1
i
a)
P (R1 U )
=
=
P (U R1 ):P (R1 )
P (U R1 ):P (R1 ) + P (U R2 ):P (R2 ) + P (U R3 )P (R3
ECON 221 HOM EW ORK 5 SU GGEST ED SOLU T ION S
1) a)T here are 9 possible assignments of construction jobs to the 3 f irms :
32 = AA; AB; AC; BA; BB; BC; CA; CB; CC and these assignments are equally likely
meaning that the probability of each of them is 1
Econ 221 - Assignment I Answer Key
September 26, 2013
1- a-The population of interest is the total number of adult residents in the
state. Since the state includes too many adult residents, we should go about
random sampling in which we choose a sample re
ECON 221 Introduction to Probability and
Statistics I
Midterm I
Solutions
Mirza Troki
c
[email protected]
Department of Economics, Bilkent University, Ankara, Turkey
Suggested solutions to the rst midterm.
October 9, 2013
1. [10 points] A random sampl
5114;4le a.
' J:
My.)
Q33
AG : Conwang
k WOVEIVL
5,6161, gamma; - I _
OLA {0995 >_ C
eé 1. 0g 0(1.M~IAI$L\:~3 ms is Dimsh
AA g NC mm
'{c A (Corwmc) OE
{Dz/miles A aw! i5
WW Ia-Ha mac mam;
As) over
OPTIDMAL
k
w_ _ _
.-"_ n
p
E
i:
beam, b?- oé
1'10". £6}.an Car (IV5
._._~
b6, :.
(aim at, [cued of 615/195? cc/KA):'!:}
Cam or a" is «Lu locus 0-? Mom Nina; mi
jjyxemobt. 0d.- iu-Ue EL o r ( uncfzvm CA/(ZK)
k: f
:3 _> (0.191 at;
EX
IdiéFam Cot/IM cm, H4,
(Marv-(4S 0; {L15
$1, _.> CHIC-(U feud.
Econ 221 - Assignment I Answer Key
September 26, 2013
1- a-The population of interest is the total number of adult residents in the
state. Since the state includes too many adult residents, we should go about
random sampling in which we choose a sample re
Econ 221 - Assignment 3 Answer Key
March 16, 2014
1)
Ri =the plane is in region i where i=1,2,3
P (Ri ) = 1=3
U=search of region 1 was unsuccessful
P (U nR1 ) = 1
i
a)
P (R1 U )
=
=
P (U R1 ):P (R1 )
P (U R1 ):P (R1 ) + P (U R2 ):P (R2 ) + P (U R3 )P (R3
ECON 221 Introduction to Statistics
Chapter 41
Continuous Random Variables
Mirza Troki
c
[email protected]
Department of Economics, Bilkent University, Ankara, Turkey
Table of Contents
1 Introduction
2
2 The Probability Distribution for a Continuous R
ECON 221 Introduction to Probability and
Statistics I
Midterm I
Solutions
Mirza Troki
c
[email protected]
Department of Economics, Bilkent University, Ankara, Turkey
Suggested solutions to the rst midterm.
October 9, 2013
1. [10 points] A random sampl
ECON 221 Introduction to Probability and
Statistics I
Midterm
Fall 2013
Mirza Troki
c
[email protected]
Department of Economics, Bilkent University, Ankara, Turkey
Instructions
You have 180 minutes. Answer every question. Allocate your time as suggest
ECON 221 Introduction to Statistics
Chapter 31
Discrete Random Variables
Mirza Troki
c
[email protected]
Department of Economics, Bilkent University, Ankara, Turkey
Table of Contents
1 Introduction
2
2 The Probability Distribution of a Discrete Random