Editing Chi-squared distribution (section)

=== Cumulative distribution function ===

[[File:Chernoff-bound.svg|thumb|400px|Chernoff bound for the [[Cumulative distribution function|CDF]] and tail (1-CDF) of a chi-squared random variable with ten degrees of freedom (<math>k = 10</math>)]]

Its [[cumulative distribution function]] is:
: <math>
    F(x;\,k) = \frac{\gamma(\frac{k}{2},\,\frac{x}{2})}{\Gamma(\frac{k}{2})} = P\left(\frac{k}{2},\,\frac{x}{2}\right),
  </math>
where <math>\gamma(s,t)</math> is the [[lower incomplete gamma function]] and <math display="inline">P(s,t)</math> is the [[Incomplete gamma function#Regularized gamma functions and Poisson random variables|regularized gamma function]].

In a special case of <math>k = 2</math> this function has the simple form:
: <math>
    F(x;\,2) = 1 - e^{-x/2}
  </math>
which can be easily derived by integrating <math>f(x;\,2)=\frac{1}{2}e^{-x/2}</math> directly. The integer recurrence of the gamma function makes it easy to compute <math>F(x;\,k)</math> for other small, even <math>k</math>.

Tables of the chi-squared cumulative distribution function are widely available and the function is included in many [[spreadsheet]]s and all [[statistical packages]].

Letting <math>z \equiv x/k</math>, [[Chernoff bound#The first step in the proof of Chernoff bounds|Chernoff bounds]] on the lower and upper tails of the CDF may be obtained.<ref>{{cite journal |last1=Dasgupta |first1=Sanjoy D. A. |last2=Gupta |first2=Anupam K. |date=January 2003 |title=An Elementary Proof of a Theorem of Johnson and Lindenstrauss |journal=Random Structures and Algorithms |volume=22 |issue=1 |pages=60–65 |doi=10.1002/rsa.10073 |s2cid=10327785 |url=http://cseweb.ucsd.edu/~dasgupta/papers/jl.pdf |access-date=2012-05-01 }}</ref> For the cases when <math>0 < z < 1</math> (which include all of the cases when this CDF is less than half):
<math style="block"> F(z k;\,k) \leq (z e^{1-z})^{k/2}.</math>

The tail bound for the cases when <math>z > 1</math>, similarly, is
: <math>
    1-F(z k;\,k) \leq (z e^{1-z})^{k/2}.
  </math>

For another [[approximation]] for the CDF modeled after the cube of a Gaussian, see under [[Noncentral chi-squared distribution#Approximation|Noncentral chi-squared distribution]].