# Chi-squared distribution

$\chi ^ {2}$- distribution

The continuous probability distribution, concentrated on the positive semi-axis $( 0, \infty )$, with density

$$p ( x) = \frac{1}{2 ^ {n / 2 } \Gamma ( {n / 2 } ) } e ^ {- {x / 2 } } x ^ { {n / 2 } - 1 } ,$$

where $\Gamma ( \alpha )$ is the gamma-function and the positive integral parameter $n$ is called the number of degrees of freedom. A "chi-squared" distribution is a special case of a gamma-distribution and has all the properties of the latter. The distribution function of a "chi-squared" distribution is an incomplete gamma-function, the characteristic function is expressed by the formula

$$\phi ( t) = \ ( 1 - 2it) ^ {-} n/2 ,$$

and the mathematical expectation and variance are $n$ and $2n$, respectively. The family of "chi-squared" distributions is closed under the operation of convolution.

The "chi-squared" distribution with $n$ degrees of freedom can be derived as the distribution of the sum $\chi _ {n} ^ {2} = X _ {1} ^ {2} + \dots + X _ {n} ^ {2}$ of the squares of independent random variables $X _ {1} \dots X _ {n}$ having identical normal distributions with mathematical expectation 0 and variance 1. This connection with a normal distribution determines the role that the "chi-squared" distribution plays in probability theory and in mathematical statistics.

Many distributions can be defined by means of the "chi-squared" distribution. For example, the distribution of the random variable $\sqrt {\chi _ {n} ^ {2} }$— the length of the random vector $( X _ {1} \dots X _ {n} )$ with independent normally-distributed components — (sometimes called a "chi" -distribution, see also the special cases of a Maxwell distribution and a Rayleigh distribution), the Student distribution, and the Fisher $F$- distribution. In mathematical statistics these distributions together with the "chi-squared" distribution describe sample distributions of various statistics of normally-distributed results of observations and are used to construct statistical interval estimators and statistical tests. A special reputation in connection with the "chi-squared" distribution has been gained by the "chi-squared" test, based on the so-called "chi-squared" statistic of E.S. Pearson.

There are detailed tables of the "chi-squared" distribution which are convenient for statistical calculations. For large $n$ one uses approximations by means of a normal distribution; for example, according to the central limit theorem, the distribution of the normalized variable $( \chi _ {n} ^ {2} - n)/ \sqrt 2n$ converges to the standard normal distribution. More accurate is the approximation

$${\mathsf P} \{ \chi _ {n} ^ {2} < x \} \rightarrow \Phi ( \sqrt 2x - \sqrt {2n- 1 } ) \ \ \textrm{ as } n \rightarrow \infty ,$$

where $\Phi ( x)$ is the standard normal distribution function.

#### References

 [1] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) [2] M.G. Kendall, A. Stuart, "The advanced theory of statistics. Distribution theory" , 1 , Griffin (1969) [3] H.O. Lancaster, "The chi-squared distribution" , Wiley (1969) [4] L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)