Transcript Lecture 1 - Lehigh University

Random Sampling
Population
 Random sample:



Independent variables
Same probability distribution
Statistics
 Point estimate

1
Examples
Mean  of a single population
2
 Variance  of a single population
 Difference in means of two population

1   2
2
Unbiasedness
An estimator  of some unknown
quantity  is said to be unbiased if the
procedure that yields  has the
property that, were it used repeatedly
the long-term average of these
estimates would be .
 That is to say, E() = .

3
Example 1
Given X = (5.8, 4.4, 8.7, 7.6, 3.1)
 The Sample Mean X
~
 The Sample Median X
 The 20% trimmed mean Xtr( 20)

4
Example 2

Sample Variance
n
S2 

2
(
X

X
)
 i
i 1
n 1
Expectation of Sample Variance
E(S )  
2
2
5
Variance of a Point Estimator
MVUE – Smallest variance estimator
 Theorem

If X1,…Xn is a random sample of size n
form a normal distribution with mean 
2
and variance  , then the sample mean X
is the MVUE for 
6
Example 1
V( X)  
(continued)
2
n
~
2
V( X)  
V( Xtr(20)
2

)
0.6n
7