Noncentral F-distribution

In probability theory and statistics, the noncentral F-distribution is a continuous probability distribution that is a noncentral generalization of the (ordinary) F-distribution. It describes the distribution of the quotient (X/n₁)/(Y/n₂), where the numerator X has a noncentral chi-squared distribution with n₁ degrees of freedom and the denominator Y has a central chi-squared distribution with n₂ degrees of freedom. It is also required that X and Y are statistically independent of each other.

It is the distribution of the test statistic in analysis of variance problems when the null hypothesis is false. The noncentral F-distribution is used to find the power function of such a test.

If ${\displaystyle X}$ is a noncentral chi-squared random variable with noncentrality parameter ${\displaystyle \lambda }$ and ${\displaystyle \nu _{1}}$ degrees of freedom, and ${\displaystyle Y}$ is a chi-squared random variable with ${\displaystyle \nu _{2}}$ degrees of freedom that is statistically independent of ${\displaystyle X}$, then

is a noncentral F-distributed random variable. The probability density function (pdf) for the noncentral F-distribution is

when ${\displaystyle f\geq 0}$ and zero otherwise. The degrees of freedom ${\displaystyle \nu _{1}}$ and ${\displaystyle \nu _{2}}$ are positive. The term ${\displaystyle B(x,y)}$ is the beta function, where

The cumulative distribution function for the noncentral F-distribution is

where ${\displaystyle I}$ is the regularized incomplete beta function.

The mean and variance of the noncentral F-distribution are