# §19.31 Probability Distributions

and occur as the expectation values, relative to a normal probability distribution in or , of the square root or reciprocal square root of a quadratic form. More generally, let () and () be real positive-definite matrices with rows and columns, and let be the eigenvalues of . If is a column vector with elements and transpose , then

19.31.1

and

§19.16(iii) shows that for the incomplete cases of and occur when and , respectively, while their complete cases occur when .

For (19.31.2) and generalizations see Carlson (1972b).