# Lecture 1 Review of probability and distributions _appendix A-C_ by dffhrtcv3

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```									Lecture 2: Review of probability and
distributions (appendix A-C)

Eco420Z
Dr. S. Chen
Outline and keywords
●   Relationship between 2 variables
●   Random sampling
●   Parameter, Estimator and estimate
●   What takes to be a good estimator?
   Small sample
   Large sample
●   Normal distribution and standard normal distribution
●   Confidence interval and hypothesis testing
●   Accuracy of an estimator: standard error and standard
deviation
Relationship between 2 variables
●   Example: Linear relationship between monthly
housing expenditure and monthly income:
Housing=164+.27 income
●   Predict the changes in housing expenditure
using changes income
   Marginal effect of income:
●   For each additional dollar of income, 27 cents are
spent on housing. Or,
●   Marginal propensity to consume = .27
●   For rich people, this may mean nothing...
●   Scale down by income:
Suppose income increases from 100 to 200:
●   Percentage point change

●   Income elasticity of housing expense
Shortcoming of linear functions
●   When income=0, housing expenditure =

●   For low levels of income, linear functions often
fail to capture the housing expenditure correctly.
Nonlinear functions
wage=5.3+.10 educ+.5 exper - .01 exper2
What is the Marginal Effect of one additional year
of work experience on wages?

What's the Percentage Point Change in wages for
one additional increase in work experience?

What is the Elasticity of wage?
●   Example2 (Quadratic log wage equation)
log(wage) =2.7+.10 educ+.2 exper -.01 exper2
   Marginal Effect of one additional year of work
experience=
   Elasticity=
●   Other two examples
   Labor supply function
hours = 33+45 log(wage)
✔   Demand function for beer
log(bottles) = 4.7+1.25 log(price)
Random sampling
●   Example: survey for UAlbany students about
their drinking behaviour
   Possible locations of interviews
   Ideal method of survey
●   Definition of random sampling:
   If {Y1,Y2,...Yn} are independent random variables
that come fro=m a common distribution, then
{Y1,Y2,...Yn} is called random sample from this
distribution. Or called independent identically
distributed (or i.i.d) random variables from that
distribution.
Parameter, estimator and estimate
●   Example (population mean and sample
average):
   Suppose we have a random sample about the
previous survey, {y1, y2, y3,y4,y5}={1,0,0,1,1}.
   Sample mean is an estimator (i.e. formula) to
approximate the population mean

   When we use the actual survey numbers, we get
the value of the sample average, also called the
estimate of the population mean
What takes to be a good estimator?
1. Unbiasedness
   An estimator is unbiased if its expectation equals the
true parameter.
   Examples of unbiased estimators
●   sample average
●   y1
●   sample variance
   Measure the bias of an estimator
bias=E[estimator] – true parameter
   Example of a biased estimator
●   Natural sample variance
2. Efficiency
   Comparing two unbiased estimators W1 and W2 for
parameter m, we say W1 is more efficient than W2 when
Var(W1)<Var(W2) for any value of m.
   Example (sample average is more efficient than the y1
estimator)
●   Mean square error
MSE= E[(W-m)2]=Var(W)+[bias(W)]2
   Take account of both unbiasedness and efficiency
●   Examples: calculate the mean and variance of the
following estimator:
   Sample average
   Y1
Sampling distribution
●   Estimators are random variable too (because
functions of random variables are random
variables). Example: sample average.
●   Thus any estimator has a distribution called the
sampling distribution.
●   Figure C.2
Large sample properties of
estimators
●   Figure C3.
   When sample size increases, the sampling
distribution would be more and more concentrated
around the true parameter.)
●   Example:
   The notorious unbiased estimator Y1 for population
mean is the widest sample distribution.
   The sample average is much more narrowly
distributed around population mean. In fact the
variance decreases whenever N increases.
●   Consistency
An estimator is called consistent estimator
 if the probability of nonzero bias decreases with sample size
 and if this probability eventually converges to zero in large
sample.
●   Sample average of a random sample must be
consistent (Law of Large Number)
●   Example:
   Sample variance is also a consistent estimator of population
variance (and also unbiased)
   Natural sample variance is also consistent (but biased).
●   Functions of consistent estimators are also
consistent.
●   Examples
   sample variance is consistent so its square root (i.e.
sample standard deviation) is a consistent estimator
of population standard deviation.
   The difference between two the sample averages is
a consistent estimator for their difference in
population means.
●   Consistency basically tells us that the
distribution of estimators are to collapse around
the true parameter as sample size gets large.
●   But this provides no information about the
shape of the distribution.
●   Asymptotic normality
   If the distribution of an estimator looks more and
more like a normal distribution as the sample size
get large, then this estimator is said to be
asymptotic normal.
Review of normal distributions
●   Suppose a random variable X is normally
distributed (i.e. has a bell shape). We often
write
X~Normal(m,s2)
to indicate that it has mean m and variance s2.
●   Standard normal distribution is a normal
distribution with zero mean and unit variance.
I.e. X~Normal(0,1).
●   Standard normal table p. 847 (can you read it?)
P{Z<a given number at the table margin)}
=number inside the table
●   Let Z be standard normal. Use the table to
P{Z<1.96}=
P{-1.96<Z}=
P{-1.96<Z<1.96}=
●   Normalization:
   i.e. demeaned by mean and rescaled by standard
deviation

   Any normal random variable Y~Normal(m,s2) can be
normalized to be standard normal.

   Example:
●   Normalization of sample average
Central Limit Theory (CLT)
●   A normalized sample average from any
random sample must be standard normal in
large sample.
●   Formally, let {y1,…yn} be a random sample with
mean m and variance s2. Then

●   Furthermore, even if we replace the population
variance in the normalization with the sample
variance, the CLT still holds.
Applications of CLT
●   Remember that sample average is a random variable.
After normalization, CLT tells us that the normalized
sample average must be standard normal.

●   This will be very useful when we construct confidence
interval for the sample average.
   Recall that P{-1.96<Z<1.96}=.95 if Z is standard normal.
   Can you construct a 95% confidence interval of sample
average for the estimation of population mean?
What if the sample is not large
enough?
●   Then the CLT cannot apply. So the normalized
sample average (using sample deviation)
cannot be standard normal.
●   Student-t
Y m
~ t n 1
s n

●   Student-t table (p. 849; can you read it?)
●   Suppose you have a small sample (n=20). Can you
construct the 95% confidence interval of sample
average for the estimation of population mean?
   the critical points (for 2 sides):
   the critical points (for 1 side)

●   If the sample is large, then use the Standard Normal
   The critical points (for 2 tails)
   the critical points (for 1 side)
Hypothesis testing
●   Example: want to test whether it’s true that
more than ½ of UAlbany students drink weekly.
   Null hypothesis:        H0:  = 0.5
   Alternative hypothesis: H1:  > 0.5 (one-sided)
●   Procedure (use confidence intervals):
   Survey and get a random sample {1,0,0,1,1}
   Estimate the sample average y = 0.6
   Construct the 95% one-sided confidence interval
using the true value (0.5):
Example: Race discrimination in
hiring (p.787)
●   Consider 5 pairs of people interview for several jobs.
In each pair, one person was black and the other is
white. Their resumes show they are virtually the same
in terms of education and experience. We observe
their outcomes for the 241 interviews. Let b and b
indicate the probability of having a job offer for black
and for white, resp.
   Construct hypotheses: H0: b- w =0; H1: b- w 0
   Calculate sample averages of the difference
B  W = 224  .357 = .133
   Calculate sample standard deviation of the diff: s=.482.
   Construct the 95% confidence interval
   Construct the 99% confidence interval
Accuracy of the sample average
Suppose we have random sample y ~ (m,s2) and its
sample average is      2
s
y ~ (m,          )
n
●   Standard deviation of sample average sd(y) = s
n
●   To estimate   s 2,   we use the sample standard variance.
n
 ( yi  y )
2

s 2 = i =1
n 1
●   We call s the standard deviation of y.
●   The unbiased estimate of sd(sample avg)is called
standard error of the sample average
sˆ
se(y) =
n
Example (Problem set C.8)
●   Larry Bird has FGA=1206 and FGM=455. The
outcome of each shot (denoted by Yi) is a
zero-one Bernoulli variable.
●   Yi =   1 with probability 
0 with probability 1-
1. To estimate  , we use the sample average
FGM/FGA=455/1206.
2. Find standard deviation of the sample average:
Given that Y is Bernoulli with mean equal to , the
variance of Y is
Var (Y ) =  (1   )

Thus the variance of the sample average is Var (Y ) =  (1   )
n
where n is FGA of a given player.
The standard deviation of the sample average is
 (1   )
sd (Y ) =
n
Note that the sample counter part of the standard
deviation is standard error
Y (1  Y )
se(Y ) =
n
3. By Central Limit Theorem, the normalized
sample average is standard normal in large
sample.
Y 
~ N (0,1)
se(Y )

Hypothesis testing for Larry Bird for the 1%
significance level:
H0: =.5
H1: >.5

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