finv

F inverse cumulative distribution function

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.

Examples

Inverse of F Distribution cdf

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) + ")")

Ratio of Sample Variances

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].

Estimate Ratio of Two Variances

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

p — Probability values at which to evaluate inverse of cdf

scalar value in [0,1] | array of scalar values

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

nu1 — Number of degrees of freedom (numerator)

positive scalar value | array of positive scalar values

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

nu2 — Number of degrees of freedom (denominator)

positive scalar value | array of positive scalar values

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

x — inverse cdf values

scalar value | array of scalar values

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

F Distribution

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

$$x=F^{-1}\left(p|{\nu }_1,{\nu }_2\right)=\left\{x:F\left(x|{\nu }_1,{\nu }_2\right)=p\right\}$$

where

$$p=F(x|{\nu }_1,{\nu }_2)={\displaystyle \underset{0}{\overset{x}{\int }}\frac{\Gamma \left[\frac{({\nu }_1+{\nu }_2)}{2}\right]}{\Gamma \left(\frac{{\nu }_1}{2}\right)\Gamma \left(\frac{{\nu }_2}{2}\right)}}{\left(\frac{{\nu }_1}{{\nu }_2}\right)}^{\frac{{\nu }_1}{2}}\frac{t^{\frac{{\nu }_1-2}{2}}}{{\left[1+\left(\frac{{\nu }_1}{{\nu }_2}\right)t\right]}^{\frac{{\nu }_1+{\nu }_2}{2}}}dt$$

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