The continuous probability distribution, concentrated on the positive semi-axis
,
with density
where
is the gamma-function and the positive integral parameter
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
and the mathematical expectation and variance are
and
,
respectively. The family of
"chi-squared"
distributions is
closed under the operation of convolution.
The
"chi-squared"
distribution with
degrees of freedom can be derived as the distribution of the sum
of the squares of independent random variables
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
— the length of the random vector
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
-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
one uses approximations by means of a normal distribution; for example, according
to the central limit theorem, the distribution of the normalized variable
converges to the standard normal distribution. More accurate is the approximation
where
is the standard normal distribution function.
See also
Non-central
"chi-squared" distribution.