This paper by Jean-Pierre Imhof presents methods for computing the distribution of quadratic forms in normal variables, which are crucial in statistical applications. The focus is on the probability \( P(Q > x) \) for a given value \( x \). The author reviews existing methods, including those by Box (1954), Gurland (1955), and Grad & Solomon (1955), and highlights their limitations, particularly in handling non-central variables. Imhof introduces a decomposition of the quadratic form \( Q \) into a sum of independent chi-square variables with non-central parameters. This decomposition is useful for both central and non-central variables. For non-central variables, the characteristic function of \( Q \) is derived, and the probability density function \( g(x) \) is obtained through numerical integration. The paper also discusses finite expressions for the probability \( P(Q > x) \) and provides a theorem for computing this probability using finite series of chi-square probabilities. Additionally, it presents numerical methods for inverting the characteristic function to obtain the cumulative distribution function \( F(x) \) of \( Q \). Finally, the accuracy of some approximations, such as Pearson's three-moment approximation, is evaluated. The results show that the three-moment approximation is more accurate than the standard two-moment (Patnaik) approximation, especially for positive quadratic forms. The paper concludes with a discussion of the computational methods and their practical applications.This paper by Jean-Pierre Imhof presents methods for computing the distribution of quadratic forms in normal variables, which are crucial in statistical applications. The focus is on the probability \( P(Q > x) \) for a given value \( x \). The author reviews existing methods, including those by Box (1954), Gurland (1955), and Grad & Solomon (1955), and highlights their limitations, particularly in handling non-central variables. Imhof introduces a decomposition of the quadratic form \( Q \) into a sum of independent chi-square variables with non-central parameters. This decomposition is useful for both central and non-central variables. For non-central variables, the characteristic function of \( Q \) is derived, and the probability density function \( g(x) \) is obtained through numerical integration. The paper also discusses finite expressions for the probability \( P(Q > x) \) and provides a theorem for computing this probability using finite series of chi-square probabilities. Additionally, it presents numerical methods for inverting the characteristic function to obtain the cumulative distribution function \( F(x) \) of \( Q \). Finally, the accuracy of some approximations, such as Pearson's three-moment approximation, is evaluated. The results show that the three-moment approximation is more accurate than the standard two-moment (Patnaik) approximation, especially for positive quadratic forms. The paper concludes with a discussion of the computational methods and their practical applications.