Chapter 3 Exercises
(1) A fair coin is tossed three times. Let X be the total number of heads that appeared in the
three tosses.
a. Write down the probability distribution of X.
b. Find P(X = 3|X 2).
(2) An urn contains five balls numbered 1 to 5. Two bal

Chapter 6: Special Distributions:
Bernoulli Random Variable: Outcomes of the trials are independent of one another, 0 < p < 1
P(success) = p; P(failure) = 1 P(success) = 1 p
No. of successes, X in a sequence of n independent trials each with constant P(su

Some Useful Mathematical Tools
Limits
1
= lim1 +
n
n
(1) e = lim(1 + t )1 / t
t 0
n
n
b
Ex. To find lim1 + for b > 0, let m = n/b, then,
n
n
b
1
b
lim1 + = lim 1 + = e b
n
m
m
n
n
m
(2) LHpitals Rule: f(x) and g(x) are two functions such that

2b)
It is a linear model because as seen from the scatter plot, data plots of volume and Y seem to fall along a straight line.
xi y i
xi
n
i=1
yi
n
i=1
x i2
n
i=1
xi
n
i=1
n
n
i=1
n
^b=
28 (368.974 ) ( 465.04)(20.16)
28 ( 8189.164 )( 465.04)2
956.0656
130

Question 1a
X Bernoulli ( p )
Let X be the number of orders with a higher price
p is the proportion of orders with higher price
H 0 : p=0.5
H 1 : p>0.5
Question 1b
Since we are investigating on the proportion and hence, probability of price
increasing and

Statistical Estimation
Hypothesis Testing
Interval Estimation
Confidence interval
- Level of confidence: a measure of our level of belief
- Margin of error: a measure of the precision of our
estimate
Quality of regression
(Residual Plots)
Correlation & Re

P B A P A
mm = “5(3)” P(AIB) = ‘ 11(g)‘ ’ P(X) = jxsx f(x)dx, PM = jysy r(y)dy
_ w _ F(X’y) co °°
P(X S x|Y 5’) — P(Y s y) _ F(y) f(x) =/ f(x,y)dy, W) =/ WWW
%_—w/
P(X S x|Y = y) = X<x f(x|y)dx = /X<x dx cf_ partition ru|e; P(.)=P(. and odd)+P(. and even)

1
Chapter 9 Exercises
1. Suppose X is a variable that follows the normal distribution with known standard
deviation = 0.3 but unknown mean .
(a) Construct a 95% confidence interval for if a random sample of n = 16 observations
of X has sample mean x = 5.

Chapter 5 Exercises
(1) A random variable X has the distribution:
X
0
1
4
6
P(X)
1
4
3
16
5
16
1
4
Find the expected value, variance, and standard deviation of X.
(2) A number Y is chosen at random from the set S = cfw_1, 0, 2. Find the expected value, va

1
Chapter 10 Exercises
q
Pn
1
)2 = 30. Assuming
1. A sample of 30 observations has x = 137 and s = n1
i=1 (xi x
the data follow a normal distribution, use these values to test whether or not the population
mean is different from 150.
2. A variable X follo

1
Chapter 11 Exercises
1. A geyser is a hot spring that periodically erupts, throwing water into the air.
Geysers are extremely rare. There are only about 50 known geyser
fields around the world. One of the largest fields is the Valley of
Geysers in Kamch

Chapter 4 Exercises
(1) Consider the following bivariate distribution:
X
P(X, Y )
1
2
3
5
1/9
1/6
0
Y
10
1/9
2/9
0
15
0
1/6
2/9
(a) Show that P(X, Y ) is a valid probability distribution function
(b) Show that X and Y are not independent
(c) Find the marg

Chapter 7 Exercises
(1) A development economist wants to study the economic situation in a remote area in Mongolia.
The following table gives the number of heads of cows (X) owned by each household in the area.
Household
1
2
3
4
5
6
Heads of cow
10
12
14

Chapter 8 Exercises
(1) A kelong is Malay for an offshore platform built
The platform is supported on wooden stilts of about
In addition to supporting the platform, the wooden stilts are also
used to construct a funnel-like structure to guide fish into th

Problem Set 5
ECON 107
Due Tuesday, September 27
p
0
1. (Bonus) Derive the asymptotic distribution for b0 . Hint: use b0 = Y X b1 and n(b1 1 ) Wan2
n
Pn
Pn
where Wn2 = 1n i=1 (Xi X )Ui ) and an = n1 i=1 (Xi X)2 . Check equation 4.22 in S&W.
2. Suppose th

Chapter 6 Exercises
A glen is a Goidelic word for a deep valley, usually with
a river that runs through it. The Angus Glens is a
collection of five glens in Scotland, UK; the glens are
popular among hikers from around the world. Arguably
the most beautifu

Probability
Probability: is the long run frequency
of an outcome.
Independent
P(A and B)=P(A)P(B)
A & B are independent means the
occurrence of one event d oes n ot
change the chance of the other.
Disjoint/Mutually exclus

3a)
f ( s )=
For
1
k s k1 es , k =2
( k 1 ) !
1
2 s 21 e s
( 21 ) !
2
s
se
=1,
3
P ( s> 3 )=1P ( s 3 )
1 ( 12 s es ) ds
10.8009
0.1991
0
For
=2,
3
P ( s> 3 )=1P ( s 3 )
1 ( 22 s e2 s ) ds
10.9826
0.01735
0
Therefore, since P ( s> 3|=1 )> P ( s>3|

1ai)
X is the number of pieces of carrot
X~Bin (20,0.1)
Y is the number of pieces of hay pellet
Y~Bin (20,0.5)
E(A)
= P(X10)(1) + P(X<10) * (1+3)
= [1 - P(X9)] + P(X9) * 4
= $3.9999785
$4
E(B)
= P(Y10)(3) + P(Y<10) * (1+3)
= [1 - P(X9)] * 3 + P(X9) * 4
=

AY2009-10 Term 2 Final Examinations (Sample)
April 2010
STAT 151 Introduction to Statistical Theory
Name of Instructor: Yang Zhenlin
INSTRUCTIONS TO CANDIDATES
1
The time allowed for this examination paper is 2 hours.
2
This examination paper contains a t

AY2009-10 Term 2 Final Examinations (Sample)
April 2010
STAT 151 Introduction to Statistical Theory
Name of Instructor: Yang Zhenlin
INSTRUCTIONS TO CANDIDATES
1
The time allowed for this examination paper is 2 hours.
2
This examination paper contains a t

ECON20003 - Quantitative Methods 2
Tutorial 7
First Semester, 2016
for week of 18 April to 22 April, 2016
PART A: Questions to be completed before the tutorial
Question A1
Consider the simple regression model
yi b0 b1 xi
which has been estimated with some

ECON20003 Quantitative Methods 2
FirstSemester,2016
Tutorial10
forweekofMay9toMay13,201
PART A
Question to be completed BEFORE the tutorial
Question A1 (Taken from First Semester 2010 Final Exam)
A chain of car repair service centres wanted to know whethe

ECON20003 Quantitative Methods 2
First Semester, 201
Tutorial 9
for week of May to May , 201
PART A
Questions to be completed BEFORE the tutorial
Question A1
The marketing manager for a chain of hardware stores needed more information
about the effectiven

ECON20003 - Quantitative Methods 2
Tutorial 4
First Semester, 2016
for week of 21 March to 25 March, 2016
PART A: Questions to be completed before the tutorial
Question A1
In a survey of instructors on quality of the new edition of a textbook compared to

ECON20003 - Quantitative Methods 2
Tutorial 3
First Semester, 2016
for week of 14 March to 18 March, 2016
PART A: Questions to be completed before the tutorial
Question A1
The risk of an investment is often measured by the variance of the return on invest