Noncentral F Distribution Calculator

Compute the probability density, lower cumulative probability, and upper-tail probability of the noncentral F distribution for given degrees of freedom d1, d2 and noncentrality λ, with graphs.

Input

Enter the degrees of freedom d1 and d2, the noncentrality λ, and an evaluation point x to compute the density, lower cumulative probability, and upper-tail probability of the noncentral F distribution.

Evaluation point x

Enter a real number of 0 or greater.

Numerator degrees of freedom d1

Enter a positive value.

Denominator degrees of freedom d2

Enter a positive value.

Noncentrality λ

Enter a value of 0 or greater. 0 gives the central F distribution.

Result

Lower cumulative probability at x=3, d1=5, d2=20, λ=4

0.81824418

Upper-tail probability

0.18175582

Density f(x)

0.14457091

Mean

Variance

1.94444444

Probability density function (PDF)

Cumulative distribution function (CDF)

How it works

The noncentral F distribution arises as the ratio of a noncentral chi-square variable (df d1, noncentrality λ) over d1 to a central chi-square variable (df d2) over d2. When λ equals 0 it reduces to the ordinary central F distribution.
The cumulative distribution function (lower-tail) is evaluated as an infinite series of regularized incomplete beta functions weighted by Poisson terms e^(−λ/2)(λ/2)^j / j!. Each term is I_y(d1/2＋j, d2/2) with y＝d1·x/(d1·x＋d2).
The probability density function uses the same Poisson-weighted series. Each term is (d1/d2)^(d1/2＋j)·x^(d1/2−1＋j)·(1＋d1·x/d2)^(−(d1＋d2)/2−j) divided by the beta function B(d2/2, d1/2＋j).
The mean is d2(d1＋λ)/(d1(d2−2)) when d2 is greater than 2. The variance is defined only when d2 is greater than 4; otherwise these values are not shown.
The noncentral F distribution is used in power analysis for ANOVA and regression as the distribution of the test statistic when the null hypothesis is false.
The gamma and incomplete beta functions are evaluated with built-in numerical methods (Lanczos approximation and continued fractions) and do not rely on external libraries. Results are approximations.

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