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# tcdf

Student's t cumulative distribution function

## Syntax

p = tcdf(x,nu)
p = tcdf(x,nu,'upper')

## Description

p = tcdf(x,nu) returns the cumulative distribution function (cdf) of the Student's t distribution at each of the values in x using the corresponding degrees of freedom in nu. x and nu can be vectors, matrices, or multidimensional arrays that all have the same size. A scalar input is expanded to a constant array with the same dimensions as the other inputs.

p = tcdf(x,nu,'upper') returns the complement of the Student's t cdf at each value in x, using an algorithm that more accurately computes the extreme upper tail probabilities.

## Examples

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### Compute Student's t cdf

mu = 1;     % Population mean
sigma = 2;  % Population standard deviation
n = 100;    % Sample size

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

xbar = mean(x);  % Sample mean
s = std(x);      % Sample standard deviation
t = (xbar - mu)/(s/sqrt(n))

t =

1.0589


p = 1-tcdf(t,n-1) % Probability of larger t-statistic

p =

0.1461



This probability is the same as the p value returned by a t test of the null hypothesis that the sample comes from a normal population with mean

[h,ptest] = ttest(x,mu,0.05,'right')

h =

0

ptest =

0.1461



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### Student's t cdf

The cumulative distribution function (cdf) of Student's t distribution is

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

where ν is the degrees of freedom and Γ( · ) is the Gamma function. The result p is the probability that a single observation from the t distribution with ν degrees of freedom will fall in the interval [–∞, x].