Stat1801
Probability and Statistics: Foundations of Actuarial Science
Fall 2010-2011
§ 6.2
Sampling Distribution of Sample Statistics
Sampling distribution
: the distribution of possible values of a sample statistic
over all random samples of a given size.
Example 6.3
Consider a hypothetical population with size
5
=
N
:
{ 2, 4, 6, 8, 10 }
If we draw a number randomly from this population and denote it as
, then the
probabilistic behaviour of
can be described by the following pmf :
1
X
1
X
()
(
)
5
1
10
8
6
4
2
1
1
1
1
1
=
=
=
=
=
=
=
=
=
=
X
P
X
P
X
P
X
P
X
P
Hence this pmf completely described the population. Moreover, it can be easily
determined that the population mean and population variance are
6
1
=
=
X
E
μ
,
( )
1
2
X
Var
=
σ
.
Suppose we draw one more number from the
same population
and denote it as
(sample with replacement). The pmf of
is the same as
and
are
independent. Then
2
X
2
X
1
X
2
1
,
X
X
{ }
2
1
,
X
X
forms a random sample with size
. From a
particular random sample, we can calculate the following sample statistics:
2
=
n
2
2
1
X
X
X
+
=
,
2
1
2
1
2
2
1
2
2
2
1
2
X
X
X
X
X
X
S
−
=
−
+
−
−
=
Since
are
random,
2
1
,
X
X
X
and
2
S
are also random. Their probabilistic
behaviour can be described by the following table:
Sample
X
−
X
2
S
2
2
−
S
Probability
2, 2
2
- 4
0
- 8
1/25
2, 4
3
- 3
2
- 6
2/25
2, 6
4
- 2
8
0
2/25
2, 8
5
- 1
18
10
2/25
2, 10
6
0
32
24
2/25
4, 4
4
- 2
0
- 8
1/25
4, 6
5
- 1
2
- 6
2/25
4, 8
6
0
8
0
2/25
4, 10
7
1
18
10
2/25
6, 6
6
0
0
- 8
1/25
p. 108