Feb 1
st
lecture
Sampling from the Normal Population
*
Example
: We wish to estimate the distribution of heights of adult US male. It is believed that the
height of adult US male follows a normal distribution
2
(
,
)
N
μ σ
Def.
Simple random sample: A sample in which every subject in the population has the same
chance to be selected.
X
:
The Random Variable denote the height of a adult male we will choose randomly from the
population
So
2
~
(
,
)
X
N
μ σ
: the distribution of a randomly selected subject is the population
distribution.
Theorem 1
Sampling from the normal population
Let
i.i.d.
2
1
2
,
,...,
~
(
,
)
n
X
X
X
N
μ σ
, where i.i.d stands for independent and identically
distributed
1.
2
~
(
,
)
X
N
n
σ
μ
2.
2
2
2
1
1
2
2
(X
)
(
1)
~
(Chi Square distribution with (
1) degrees of freedom),
n
i
i
n
X
n
S
n
χ
σ
σ
=



=

∑
*Reminder: The Sample variance
2
S
is defined as:
2
2
1
(X
)
1
n
i
i
X
S
n
=

=

∑
*Def 1: The Chisquare distribution is a special gamma distribution (*** Please find out
which one it is.)
*Def 2: Let
i.i.d.
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 Spring '08
 Zhu,W
 Normal Distribution, Chisquare distribution, Chi Square Distribution, p.d.f. f X1

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