ECON 503
Lecture Notes 20
Neslihan Uler
1
Type I and Type II errors
Type I error: Rejecting H0 when H0 is true
Type II error: Fail to reject H0 when H0 is false
Remember that the probability of committing a Type I error is referred to
as a tests level of
ECON 503
Lecture Notes 10
Neslihan Uler
1
Independence
Definition
Two random variables X and Y are said to be independent if for every
interval A and every interval B,
P( X A, Y B) = P( X A) P(Y B)
2
Independence
Theorem
The random variables X and Y are i
ECON 503
Lecture Notes 11
Neslihan Uler
1
Mean and Variance for functions of random variables
Theorem
Suppose X and Y are discrete random variables, with joint pdf pX,Y(x,y)
and let g(X,Y) be a function of X and Y. Then, the expected value of
the random v
ECON 503
Lecture Notes 9
Neslihan Uler
1
Marginal pdfs for continuous random variables
Theorem
Suppose X and Y are jointly continuous with joint pdf f X ,Y ( x, y )
Then the marginal pdfs, f X (x) and fY ( y ) , are given by
f X ( x) =
f
X ,Y
( x, y )dy a
ECON 503
Lecture Notes 13
Neslihan Uler
1
Poisson Limit
= =
(1 ) , = 0,1,2, ,
Theorem: Suppose X is a binomial random variable, where
If and 0 in such a way that = remains constant, then
()
lim
( = ) =
,0,
!
Note: Even though this is an asymptotic r
ECON 503
Lecture Notes 18
Neslihan Uler
1
Properties of Estimators
Unbiasedness
Efficiency
Sufficiency (not included in the exam)
Consistency
2
Unbiasedness
Definition
Suppose that Y1, Y2, . , Yn is a random sample from continuous pdf
fY(y;), where is an
ECON 503
Lecture Notes 23
Neslihan Uler
1
Statistical Inference About Means with Two Populations
Suppose the data consist of two independent random samples:
X 1 , X 2 , , X n and Y1 , Y2 , , Ym . Two populations are normally distributed.
H0 : X = Y
2
2
Homework 1
(Unless otherwise mentioned, homeworks are from your book, Larsen and Marx and they are due on
Wednesdays.)
2.2.11
2.2.29
2.2.40
2.3.16
2.4.6
2.4.21
2.4.36
2.4.41
2.5.7
2.5.26
Bonus Question:
From the class web page (ctools.umich.edu> resources
ECON 503
Lecture Notes 7
Neslihan Uler
1
Continuous Random variables
Definition
A random variable that can take on any value in an interval is called
continuous.
Continuous sample space:
Contains an uncountably infinite number of outcomes
It is not poss
ECON 503:
Probability and Mathematical
Statistics
Lecture Notes 1
Neslihan Uler
1
Lecture 1
Introduction
Statistics deals with the collection and analysis of data
Descriptive
statistics
Uses numerical or graphical
methods to summarize the data
set and pro
ECON 503
Lecture Notes 5
Neslihan Uler
1
Discrete Random Variables
Definition
Given a random experiment with an outcome space S, a function Y that
assigns one and only one real number Y(s) = y to each element s in S is
called a random variable.
Definition
ECON 503
Lecture Notes 17
Neslihan Uler
1
Confidence Intervals for Proportions (The Binomial Parameter, p)
Let X be a binomial random variable defined on n independent trials
for which p=P(success).
X np
np (1 p )
X /n p
p (1 p )
n
~
N (0,1)
~
for large n
ECON 503
Lecture Notes 22
Neslihan Uler
1
Drawing Inferences about 2
S
1
(Yi Y ) 2 is an unbiased estimator for 2
n 1
2
Theorem
(n 1) S 2
=
2
1
2
(Yi Y ) 2 has a chi-square distribution with n-1
degrees of freedom.
2
Motivation
3
Theorem (a)
A 100(1-)% co
ECON 503
Lecture Notes 21
Neslihan Uler
1
Student t distribution
Suppose 1 , 2 , , is a random sample from a normal distribution with mean
and standard deviation .
If the true variance is known, then Z =
/
has a normal distribution.
We can use the unb
Econ 503
Homework 3
3.2.8
3.2.15
3.2.28
3.3.14
3.5.6
3.5.36
3.6.1 (solve this question by using the definition of variancedo not use the formula for this question.)
3.4.6
Bonus question: Use Stata to do the following problems. Submit a log-file with your
Homework 2
2.6.14
2.6.23
2.6.36
2.6.57
2.7.20
3.3.3
3.5.22
Let Y be a discrete random variable with mean and standard deviation . If a and b are constants,
prove that V(aY+b) = a2 2.
Bonus Question:
Use 503class.dta and Stata program to do the following e
ECON 503
Lecture Notes 2
Neslihan Uler
1
Probability
Lecture 2
P(A) : Probability of event A
P : Probability function
Definition
Suppose S is a sample space associated with an experiment. To every
event A defined over S, we assign a number, P(A), called t
ECON 503
Lecture Notes 4
Neslihan Uler
1
Counting
If a job consists of k separate tasks, the i th of which can be done in
ways, i=1, ,k, then the entire job can be done in n1 n2 nk
ways.
Theorem
Example
A balanced coin is tossed six times. What is the to
ECON 503
Lecture Notes 3
Neslihan Uler
1
Conditional probability
Definition
The conditional probability of an event A, given that an event B has
occurred is equal to P( A | B) = P( A B)
P( B)
Provided P( B) > 0
2
Conditional probability
Exercise
Suppose a
Econ 503
Homework 8
1) 4.2.22
2) A certain type of tree has seedlings randomly dispersed in a large area, with the mean density of
seedlings being approximately five per square yard. If a forester randomly and independently locates
two 1-square yard sampl
Homework 11
For practice only. This homework will not be graded.
7.4.2
7.4.5
7.4.12
7.4.19
7.5.3
7.5.8
Stata Question:
Use 503class.dta to do the following exercise.
1. Test the null hypothesis whether height = 160cm for female students against height>160
Discussion, ECON 503
Gail Lucasan
September 17, 2015
Some things we learned this week:
The outcome space (S) is the set of all possible outcomes of a given experiment.
Probability is a function P that is dened over all subsets of S that has the followin
*Stata code for Week 6
set obs 5000
gen x = rnormal(10.5,2)
gen z = (x-10.5)/2
hist x, normal name(histx)
hist z, normal name(histz)
graph combine histx histz
clear all
set obs 5000
foreach n of numlist 1 2 5 10 50 100 1000cfw_
gen x`n'= rbinomial(`n',0.1
Discussion, ECON 503
Gail Lucasan
November 6, 2015
Review of standard distributions
Distribution
X Binomial(n,p)
p (0, 1)
p.d.f.
p(x) = n px (1 p)1x , x cfw_0, 1, 2, .
x
E[X] = np, V (X) = np(1 p)
X Poisson()
p(x) =
>0
X Uniform(a,b)
a<b
X N (, 2 )
>0
X E
Discussion Answers, ECON 503
Gail Lucasan
November 6, 2015
1. Since X is Binomial, its m.g.f. is M (t) = (1 0.2 + 0.2et )15 . For a proof of this, see
Lecture Notes 12.
2.
mY (t) = E[etY ] = E[et(aX+b) ] = E[etb etaX ] = etb E[taX ] = mX (ta).
3.
M (t) =
Discussion, ECON 503
Gail Lucasan
November 13, 2015
Poisson distribution
A Poisson r.v. is one that counts the number of changes that occur in a given
continuous interval, assuming that the changes are independent.
The p.m.f. has the form
f (x) =
x e
,
*Week 8 Stata exercise (11/13/2015)
/*
Let cfw_Xi, n=25 be i.i.d. random variables with Xi ~ Unif[0,1].
Show that the distribution of Xbar and X1+X2+.+X25 is approximately normal.
*/
clear all
set obs 5000
*First simulate the realizations or values of th
Discussion, ECON 503
Gail Lucasan
November 20, 2015
Gamma distribution
A gamma r.v. has the following p.d.f.
f (y) =
r r1 y
y e
,y > 0
(r)
where > 0, r > 0, and (r) =
r1 y
e
0 y
dy is the gamma function.
For this class, we focus on when r is an integer
Section 10
Thursday, Nov 10th
Topics covered:
Finite sample properties of estimators: unbiasedness; e ciency
Large sample properties of estimators: consistency
Problem 1
(based on problem 5.5.4, pg. 323 from the textbook, but I have
extended it considerab
Moments of Random Variables (Contd)
September 30, 2016
Example
Consider two discrete random variables: X with support
f 2, 1, 0, 1, 2g, each with probability 15 , Y with support
f 6, 3, 0, 3, 6g, each with probability 15 .
Variance of continuous random va
Lecture 14-15: Multivariate distributions
Focus: bivariate case (2 variables)
I. Discrete case
Denition 1 The joint PDF of two variable X and Y is given by: fX;Y (x; y) =
P (X = x; Y = y)
Example 1
A fair coin is tossed three times.
Let
X=number of heads
ASYMPROTIC (LARGE SAMPLE)
PROPERTIES OF ESTIMATORS
Consistency. Consistency of the sample mean/Law of Large
Numbers (LLN) (Chapter 4, section 4.3 and Chapter 5, example
5.7.2)
Asymptotic normality. Central Limit Theorem (CLT) (Chapter 4, section 4.3)
The