Chi-Squared Distribution
المؤلف:
Abramowitz, M. and Stegun, I. A
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
4-4-2021
4024
Chi-Squared Distribution
If 
have normal independent distributions with mean 0 and variance 1, then
 |
(1)
|
is distributed as 
with 
degrees of freedom. This makes a 
distribution a gamma distribution with 
and 
, where 
is the number of degrees of freedom.
More generally, if 
are independently distributed according to a 
distribution with 
, 
, ..., 
degrees of freedom, then
 |
(2)
|
is distributed according to 
with 
degrees of freedom.
The probability density function for the 
distribution with 
degrees of freedom is given by
 |
(3)
|
for 
, where 
is a gamma function. The cumulative distribution function is then
where 
is an incomplete gamma function and 
is a regularized gamma function.
The chi-squared distribution is implemented in the Wolfram Language as ChiSquareDistribution[n].
For 
, 
is monotonic decreasing, but for 
, it has a maximum at
 |
(8)
|
where
 |
(9)
|
The 
th raw moment for a distribution with 
degrees of freedom is
giving the first few as
The 
th central moment is given by
 |
(16)
|
where 
is a confluent hypergeometric function of the second kind, giving the first few as
The cumulants can be found via the characteristic function
Taking the natural logarithm of both sides gives
 |
(23)
|
But this is simply a Mercator series
 |
(24)
|
with 
, so from the definition of cumulants, it follows that
 |
(25)
|
giving the result
 |
(26)
|
The first few are therefore
The moment-generating function of the 
distribution is
so
If the mean is not equal to zero, a more general distribution known as the noncentral chi-squared distribution results. In particular, if 
are independent variates with a normal distribution having means 
and variances 
for 
, ..., 
, then
 |
(42)
|
obeys a gamma distribution with 
, i.e.,
 |
(43)
|
where 
.
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 940-943, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 535, 1987.
Kenney, J. F. and Keeping, E. S. "The Chi-Square Distribution." §5.3 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 98-100, 1951.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 209-214, 1992.
Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 115-116, 1992.
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