# tinv

Student's t inverse cumulative distribution function

## Description

example

x = tinv(p,nu) returns the inverse cumulative distribution function (icdf) of the Student's t distribution evaluated at the probability values in p using the corresponding degrees of freedom in nu.

## Examples

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Find the 95th percentile of the Student's t distribution with 50 degrees of freedom.

p = .95;
nu = 50;
x = tinv(p,nu)
x = 1.6759

Compute the 99th percentile of the Student's t distribution for 1 to 6 degrees of freedom.

percentile = tinv(0.99,1:6)
percentile = 1×6

31.8205    6.9646    4.5407    3.7469    3.3649    3.1427

Find a 95% confidence interval estimating the mean of a population by using tinv.

Generate a random sample of size 100 drawn from a normal population with mean 10 and standard deviation 2.

mu = 10;
sigma = 2;
n = 100;

rng default   % For reproducibility
x = normrnd(mu,sigma,n,1);

Compute the sample mean, standard error, and degrees of freedom.

xbar = mean(x);
se = std(x)/sqrt(n);
nu = n - 1;

Find the upper and lower confidence bounds for the 95% confidence interval.

conf = 0.95;
alpha = 1 - conf;
pLo = alpha/2;
pUp = 1 - alpha/2;

Compute the critical values for the confidence bounds.

crit = tinv([pLo pUp], nu);

Determine the confidence interval for the population mean.

ci = xbar + crit*se
ci = 1×2

9.7849   10.7075

This confidence interval is the same as the ci value returned by a t test of a null hypothesis that the sample comes from a normal population with mean mu.

[h,p,ci2] = ttest(x,mu,'Alpha',alpha);
ci2
ci2 = 2×1

9.7849
10.7075

## Input Arguments

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Probability values at which to evaluate the 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 nu using an array.

If either or both of the input arguments p and nu are arrays, then the array sizes must be the same. In this case, tinv expands each scalar input into a constant array of the same size as the array inputs. Each element in x is the icdf value of the distribution specified by the corresponding element in nu, evaluated at the corresponding probability in p.

Example: [0.1 0.5 0.9]

Data Types: single | double

Degrees of freedom for the Student's t distribution, 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 nu using an array.

If either or both of the input arguments p and nu are arrays, then the array sizes must be the same. In this case, tinv expands each scalar input into a constant array of the same size as the array inputs. Each element in x is the icdf value of the distribution specified by the corresponding element in nu, evaluated at the corresponding probability in p.

Example: [9 19 49 99]

Data Types: single | double

## Output Arguments

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icdf 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 and nu after any necessary scalar expansion. Each element in x is the icdf value of the distribution specified by the corresponding element in nu, evaluated at the corresponding probability in p.

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### Student’s t icdf

The Student's t distribution is a one-parameter family of curves. The parameter ν is the degrees of freedom. The Student's t distribution has zero mean.

The t inverse function is defined in terms of the Student's t cdf as

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

where

$p=F\left(x|\nu \right)={\int }_{-\infty }^{x}\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu }{2}\right)}\frac{1}{\sqrt{\nu \pi }}\frac{1}{{\left(1+\frac{{t}^{2}}{\nu }\right)}^{\frac{\nu +1}{2}}}dt,$

ν is 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.

## Alternative Functionality

• tinv is a function specific to the Student's t 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 tinv is faster than the generic function icdf.