Main Content

finv

F inverse cumulative distribution function

collapse all in page

    Syntax

    x = finv(p,nu1,nu2)

    Description

    x = finv(p,nu1,nu2) returns the inverse cumulative distribution function (icdf) of the F distribution with degrees of freedom nu1 (numerator) and nu2 (denominator), evaluated at the probability values in p.

    example

    Examples

    collapse all

    Open Live Script

    Compute the inverse cdf values evaluated at the probability values in p for an F distribution with degrees of freedom nu1 and nu2. 

    p = linspace(0.005,0.995,100);
nu1 = 8;
nu2 = 9;
x = finv(p,nu1,nu2);

    Plot the inverse cdf.

    plot(p,x)
grid on
xlabel("p")
ylabel("x = F^{-1}(p| nu1 = " + num2str(nu1)...
    + ", nu2 = " + num2str(nu2) + ")")

    Figure contains an axes object. The axes object with xlabel p, ylabel x = blank F toThePowerOf - 1 baseline (p| blank nu 1 blank = blank 8 , blank nu 2 blank = blank 9 ) contains an object of type line.

    Open Live Script

    Consider two independent random samples of size n1 and n2 drawn from normal distributions. The variance ratio of the samples has an F distribution with n1–1 and n2–1 degrees of freedom. Use the inverse cdf of the F distribution to calculate a range [0 r95] so that the variance ratio has a 95% probability of being in this range. 

    rng default % For reproducibility
n1 = 100;
n2 = 105;
p = 0.95;
r = finv([0 p],n1-1,n2-1)
    r = 1×2

         0    1.3874

    Generate two random samples from the standard normal distribution and calculate their variance ratio.

    s1 = randn([n1 1]);
s2 = randn([n2 1]);
r12 = var(s1)/var(s2)
    r12 = 
1.3749

    The variance ratio r12 is in the range [0 r95].

    Open Live Script

    Consider two independent random samples of size n1 and n2 drawn from normal populations with unknown variances var1 and var2. The samples have variances v1 and v2. Use the inverse cdf of the F distribution to calculate a 95% confidence interval for the ratio var1/var2.

    Enter the sample sizes and variances, and calculate the ratio of sample variances.

    n1 = 122;
n2 = 124;
v1 = 1.3;
v2 = 1.2;
r = v1/v2
    r = 
1.0833

    The ratio of sample variances is r.

    Compute the 95% confidence interval for the population variance ratio var1/var2.

    pCI = 95;
p = (1+pCI/100)/2;
rLow = v1/v2/finv(p,n1-1,n2-1)
    rLow = 
0.7586

    rHigh = v1/v2*finv(p,n2-1,n1-1)
    rHigh = 
1.5479

    The probability that the population variance ratio var1/var2 is in the range [rLow rHigh] is 0.95.

    Input Arguments

    collapse all

    Probability values at which to evaluate the inverse of the cdf (icdf), specified as a scalar value or an array of scalar values, where each element is in the range [0,1].

    • To evaluate the icdf at multiple values, specify p using an array.

    • To evaluate the icdfs of multiple distributions, specify nu1 and nu2 using arrays.

    If one or more of the input arguments p, nu1, and nu2 are arrays, then the array sizes must be the same. In this case, finv expands each scalar input into a constant array of the same size as the array inputs.

    Example: [0.1,0.5,0.9]

    Data Types: single | double

    Number of degrees of freedom in the numerator of the F distribution function, specified as a positive scalar value or an array of positive scalar values.

    • To evaluate the icdf at multiple values, specify p using an array.

    • To evaluate the icdfs of multiple distributions, specify nu1 and nu2 using arrays.

    If one or more of the input arguments p, nu1, and nu2 are arrays, then the array sizes must be the same. In this case, finv expands each scalar input into a constant array of the same size as the array inputs.

    Example: [ 8 7 9]

    Data Types: single | double

    Number of degrees of freedom in the denominator of the F distribution function, specified as a positive scalar value or an array of positive scalar values.

    • To evaluate the icdf at multiple values, specify p using an array.

    • To evaluate the icdfs of multiple distributions, specify nu1 and nu2 using arrays.

    If one or more of the input arguments p, nu1, and nu2 are arrays, then the array sizes must be the same. In this case, finv expands each scalar input into a constant array of the same size as the array inputs.

    Example: [ 7 6 10]

    Data Types: single | double

    Output Arguments

    collapse all

    inverse cdf values evaluated at the probabilities in p, returned as a scalar value or an array of scalar values. x is the same size as p, nu1, and nu2 after any necessary scalar expansion. Each element in x is the inverse cdf value of the distribution specified by the corresponding elements in nu1 and nu2, evaluated at the corresponding probability in p

    More About

    collapse all

    F Distribution

    The F inverse function is defined in terms of the F cumulative distribution function (cdf) as

    x=F−1(p|ν1,ν2)={x:F(x|ν1,ν2)=p}

    where

    p=F(x|ν1,ν2)=∫0xΓ[(ν1+ν2)2]Γ(ν12)Γ(ν22)(ν1ν2)ν12tν1−22[1+(ν1ν2)t]ν1+ν22dt

    The ν values are the degrees of freedom, and Γ( · ) is the Gamma function. The result x is the solution of the integral equation where you supply the probability p.

    For more information, see F Distribution.

    Alternative Functionality

    • finv is a function specific to the F distribution. Statistics and Machine Learning Toolbox™ also offers the generic function icdf, which supports various probability distributions. To use icdf, specify the probability distribution name and its parameters. Note that the distribution-specific function finv is faster than the generic function icdf.

    References

    [1] Abramowitz, M., and I. A. Stegun. Handbook of Mathematical Functions. New York: Dover, 1964.

    [2] Freund, John E. Mathematical Statistics Fifth Edition. Englewood Cliffs, NJ: Prentice Hall College Division, 1992.

    Extended Capabilities

    C/C++ Code Generation
    Generate C and C++ code using MATLAB® Coder™.

    GPU Arrays
    Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

    Version History

    Introduced before R2006a

    See Also

    | | | | |