•
Let X be a random variable with mean
𝜇
and variance
σ
. Find the
expected value and variances of random variables
•
3X
•
2X+5
Example: Sample Mean & Variance Are Unbiased1
•
X
is a random variable with mean
μ
and variance
σ
2
.
Let
X
1
,
X
2
,…,
X
n
be a
random sample of size
n
.
•
Show that the sample mean (
X
bar) is
an unbiased estimator of
μ
.
Show that the sample variance (
S
2
) is an unbiased estimator of
σ
2
.
Minimum Variance Unbiased Estimators
•
If we consider all unbiased estimators of
θ
, the one with the
smallest variance is called the
minimum variance unbiased
estimator
(MVUE).
•
If
X
1
,
X
2
,…,
X
n
is a random sample of size
n
from a normal
distribution with mean
μ
and variance
σ
2
, then the sample
X

bar is the MVUE for
μ
.
WHY?
Standard Error of an Estimator
Mean Squared Error
Conclusion:
The mean squared error (MSE) of
the estimator is equal to the variance of the
estimator plus the bias squared.
Relative Efficiency
•
The MSE is an important criterion for comparing two estimators.
•
If the relative efficiency is less than 1, we conclude that the 1
st
estimator is superior than the 2
nd
estimator.
Optimal Estimator
•
A biased estimator can be
preferred than an unbiased
estimator if it has a smaller
MSE.
•
Biased estimators are
occasionally used in linear
regression.
•
An estimator whose MSE is
smaller than that of any
other estimator is called an
optimal estimator.
Figure 78
A biased estimator
that
has a smaller variance than the
unbiased estimator
.
Example5
Suppose that the random variable
X
has the continuous uniform distribution
Suppose that a random sample of
n
= 12 observations is selected from this distribution. What is the
approximate probability distribution of
X
− 6
?
Find the mean and variance of this quantity.
Exmple5
Example6
A computer software package calculated some numerical summaries of a sample of
data. The results are displayed here:
(a) Fill in the missing quantities.
(b) What is the estimate of the mean of the population from which this sample was
drawn?
Example7
Let
X
1 and
X
2 be independent random variables with mean μ and variance σ2.
Suppose that we have two estimators
of
μ:
•
(a) Are both estimators unbiased estimators of μ?
•
(b) What is the variance of each estimator?
Example7
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 Summer '20
 Normal Distribution, Standard Deviation, Variance, Gözdem Dural Selçuk