A
distribution is a Gamma Distribution with
and
, where
is the
number of Degrees of Freedom. If
have Normal
Independent distributions with Mean 0 and Variance 1, then
![\begin{displaymath}
\chi^2\equiv \sum_{i=1}^r {Y_i}^2
\end{displaymath}](c1_1498.gif) |
(1) |
is distributed as
with
Degrees of Freedom. If
are independently
distributed according to a
distribution with
,
, ...,
Degrees of Freedom, then
![\begin{displaymath}
\sum_{j=1}^k {\chi_j}^2
\end{displaymath}](c1_1503.gif) |
(2) |
is distributed according to
with
Degrees of Freedom.
![\begin{displaymath}
P_r(x) = \cases{
{x^{r/2-1}e^{-x/2}\over\Gamma({\textstyle{...
...) 2^{r/2}} & for $0 \leq x < \infty$\cr
0 & for $x < 0$.\cr}
\end{displaymath}](c1_1505.gif) |
(3) |
The cumulative distribution function is then
where
is a Regularized Gamma Function. The Confidence Intervals can be
found by finding the value of
for which
equals a given value. The Moment-Generating Function of the
distribution is
so
The
th Moment about zero for a distribution with
Degrees of Freedom is
![\begin{displaymath}
m_n'=2^n{\Gamma(n+{\textstyle{1\over 2}}r)\over\Gamma({\textstyle{1\over 2}}r)} = r(r+2)\cdots(r+2n-2),
\end{displaymath}](c1_1523.gif) |
(13) |
and the moments about the Mean are
The
th Cumulant is
![\begin{displaymath}
\kappa_n=2^n\Gamma(n) ({\textstyle{1\over 2}}r)=2^{n-1}(n-1)! r.
\end{displaymath}](c1_1527.gif) |
(17) |
The Moment-Generating Function is
As
,
![\begin{displaymath}
\lim_{r\to\infty}M(t)=e^{t^2/2},
\end{displaymath}](c1_1532.gif) |
(19) |
so for large
,
![\begin{displaymath}
\sqrt{2\chi^2}=\sqrt{\sum_i {(x_i-\mu_i)^2\over{\sigma_i}^2}}
\end{displaymath}](c1_1533.gif) |
(20) |
is approximately a Gaussian Distribution with Mean
and Variance
. Fisher
showed that
![\begin{displaymath}
{\chi^2-r\over\sqrt{2r-1}}
\end{displaymath}](c1_1535.gif) |
(21) |
is an improved estimate for moderate
. Wilson and Hilferty showed that
![\begin{displaymath}
\left({\chi^2\over r}\right)^{1/3}
\end{displaymath}](c1_1536.gif) |
(22) |
is a nearly Gaussian Distribution with Mean
and Variance
.
In a Gaussian Distribution,
![\begin{displaymath}
P(x)\,dx={1\over\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/2\sigma^2}\,dx,
\end{displaymath}](c1_1539.gif) |
(23) |
let
![\begin{displaymath}
z \equiv (x-\mu )^2/\sigma^2.
\end{displaymath}](c1_1540.gif) |
(24) |
Then
![\begin{displaymath}
dz = {2(x-\mu)\over \sigma^2} \,dx = {2\sqrt{z}\over \sigma} \,dx
\end{displaymath}](c1_1541.gif) |
(25) |
so
![\begin{displaymath}
dx={\sigma\over 2\sqrt{z}} dz.
\end{displaymath}](c1_1542.gif) |
(26) |
But
![\begin{displaymath}
P(z)\,dz=2 P(x)\,dx,
\end{displaymath}](c1_1543.gif) |
(27) |
so
![\begin{displaymath}
P(x)\,dx=2\,{1\over \sigma\sqrt{2\pi}} e^{-z/2}\,dz = {1\over \sigma\sqrt{\pi}}
e^{-z/2}\,dz.
\end{displaymath}](c1_1544.gif) |
(28) |
This is a
distribution with
, since
![\begin{displaymath}
P(z)\,dz={z^{1/2-1}e^{-z/2}\over \Gamma({1\over 2})2^{1/2}}\,dz = {x^{-1/2}e^{-1/2}\over \sqrt{2\pi}}\,dz.
\end{displaymath}](c1_1546.gif) |
(29) |
If
are independent variates with a Normal Distribution having Means
and
Variances
for
, ...,
, then
![\begin{displaymath}
{\textstyle{1\over 2}}\chi^2\equiv \sum_{i=1}^n {(x_i-\mu_i)^2\over 2{\sigma_i}^2}
\end{displaymath}](c1_1548.gif) |
(30) |
is a Gamma Distribution variate with
,
![\begin{displaymath}
P({\textstyle{1\over 2}}\chi^2)\,d({\textstyle{1\over 2}}\ch...
...{1\over 2}}\chi^2)^{(n/2)-1}\,d({\textstyle{1\over 2}}\chi^2).
\end{displaymath}](c1_1550.gif) |
(31) |
The noncentral chi-squared distribution is given by
![\begin{displaymath}
P(x)= 2^{-n/2}e^{-(\lambda+x)/2}x^{n/2-1}F({\textstyle{1\over 2}}n, {\textstyle{1\over 4}}\lambda x),
\end{displaymath}](c1_1551.gif) |
(32) |
where
![\begin{displaymath}
F(a,z)\equiv {{}_0F_1(; a; z)\over \Gamma(a)},
\end{displaymath}](c1_1552.gif) |
(33) |
is the Confluent Hypergeometric Limit Function and
is the Gamma Function. The Mean,
Variance, Skewness, and Kurtosis are
See also Chi Distribution, Snedecor's F-Distribution
References
Abramowitz, M. and Stegun, C. 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.
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.
© 1996-9 Eric W. Weisstein
1999-05-26