In probability theory and statistics, the generalized chi-squared distribution (or generalized chi-square distribution) is the distribution of a quadratic function of a multinormal variable (normal vector), or a linear combination of different normal variables and squares of normal variables. Equivalently, it is also a linear sum of independent noncentral chi-square variables and a normal variable. There are several other such generalizations for which the same term is sometimes used; some of them are special cases of the family discussed here, for example the gamma distribution.

The generalized chi-squared variable may be described in multiple ways. One is to write it as a weighted sum of independent noncentral chi-square variables ${\displaystyle {{\chi }'}^{2}}$ and a standard normal variable ${\displaystyle z}$:

Here the parameters are the weights ${\displaystyle w_{i}}$, the degrees of freedom ${\displaystyle k_{i}}$ and non-centralities ${\displaystyle \lambda _{i}}$ of the constituent non-central chi-squares, and the coefficients ${\displaystyle s}$ and ${\displaystyle m}$ of the normal. Some important special cases of this have all weights ${\displaystyle w_{i}}$ of the same sign, or have central chi-squared components, or omit the normal term.

Since a non-central chi-squared variable is a sum of squares of normal variables with different means, the generalized chi-square variable is also defined as a sum of squares of independent normal variables, plus an independent normal variable: that is, a quadratic in normal variables.

Another equivalent way is to formulate it as a quadratic form of a normal vector ${\displaystyle {\boldsymbol {x}}}$:

Here ${\displaystyle \mathbf {Q_{2}} }$ is a matrix, ${\displaystyle {\boldsymbol {q_{1}}}}$ is a vector, and ${\displaystyle q_{0}}$ is a scalar. These, together with the mean ${\displaystyle {\boldsymbol {\mu }}}$ and covariance matrix ${\displaystyle \mathbf {\Sigma } }$ of the normal vector ${\displaystyle {\boldsymbol {x}}}$, parameterize the distribution.

For the most general case, a reduction towards a common standard form can be made by using a representation of the following form:

where D is a diagonal matrix and where x represents a vector of uncorrelated standard normal random variables.