Sample Variance Distribution
المؤلف:
Kenney, J. F. and Keeping, E. S
المصدر:
Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.
الجزء والصفحة:
...
24-2-2021
1977
Sample Variance Distribution
Let 
samples be taken from a population with central moments 
. The sample variance 
is then given by
 |
(1)
|
where 
is the sample mean.
The expected value of 
for a sample size 
is then given by
 |
(2)
|
Similarly, the expected variance of the sample variance is given by
(Kenney and Keeping 1951, p. 164; Rose and Smith 2002, p. 264).
The algebra of deriving equation (4) by hand is rather tedious, but can be performed as follows. Begin by noting that
 |
(5)
|
so
 |
(6)
|
The value of 
is already known from equation (◇), so it remains only to find 
. The algebra is simplified considerably by immediately transforming variables to 
and performing computations with respect to these central variables. Since the variance does not depend on the mean 
of the underlying distribution, the result obtained using the transformed variables will give an identical result while immediately eliminating expectation values of sums of terms containing odd powers of 
(which equal 0). To determine 
, expand equation (6) to obtain
Working on the first term of (10),
The second term of (◇) is given by
and the third term by
Plugging (◇)-(19) into (◇) then gives
(Kenney and Keeping 1951, p. 164). Plugging (◇) and (23) into (◇) then gives
as before.
For a normal distribution, 
and 
, so
The third ane fourth moments of 
are given by
giving the skewness and kurtosis excess of the distribution of the 
as
as computed by Student. Student also conjectured that the underlying distribution is Pearson type III distribution
 |
(32)
|
where 
is the gamma function--a conjecture that was subsequently proven by R. A. Fisher. Curves are illustrated above for 
and 
varying from 
to 10.
REFERENCES:
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.
Rose, C. and Smith, M. D. Mathematical Statistics with Mathematica. New York: Springer-Verlag, 2002.
الاكثر قراءة في الاحتمالات و الاحصاء
اخر الاخبار
اخبار العتبة العباسية المقدسة
