# tcdf

Student's t cumulative distribution function

## Syntax

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

## Description

example

````p = tcdf(x,nu)` returns the cumulative distribution function (cdf) of the Student's t distribution with `nu` degrees of freedom, evaluated at the values in `x`.```

example

````p = tcdf(x,nu,'upper')` returns the complement of the cdf, evaluated at the values in `x` with `nu` degrees of freedom, using an algorithm that more accurately computes the extreme upper-tail probabilities than subtracting the lower tail value from 1.```

## Examples

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Generate a random sample of size `100` from a normally distributed population with mean `1` and standard deviation `2`.

```rng default % For reproducibility mu = 1; n = 100; sigma = 2; x = normrnd(mu,sigma,n,1);```

Compute the sample mean, sample standard deviation, and t-score of the sample.

```xbar = mean(x); s = std(x); t = (xbar-mu)/(s/sqrt(n))```
```t = 1.0589 ```

Use `tcdf` to compute the probability of a sample of size `100` having a larger t-score than the t-score of the sample.

`p = 1-tcdf(t,n-1)`
```p = 0.1461 ```

This probability is the same as the p value returned by a t test with null hypothesis that the sample comes from a normal population with mean `1` and alternative hypothesis that the mean is greater than `1`.

```[h,ptest] = ttest(x,mu,0.05,'right'); ptest```
```ptest = 0.1461 ```

Determine the probability that an observation from the Student's t distribution with degrees of freedom `99` falls on the interval `[10 Inf]`.

`p1 = 1 - tcdf(10,99)`
```p1 = 0 ```

`tcdf(10,99)` is nearly `1`, so `p1` becomes `0`. Specify `'upper'` so that `tcdf` computes the extreme upper-tail probabilities more accurately.

`p2 = tcdf(10,99,'upper')`
```p2 = 5.4699e-17 ```

You can also use `'upper'` to compute a right-tailed p-value.

## Input Arguments

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Values at which to evaluate the cdf, specified as a scalar value or an array of scalar values.

• To evaluate the cdf at multiple values, specify `x` using an array.

• To evaluate the cdfs of multiple distributions, specify `nu` using an array.

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

Example: `[-1,0,3,4]`

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 cdf at multiple values, specify `x` using an array.

• To evaluate the cdfs of multiple distributions, specify `nu` using an array.

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

Example: `[9,19,49,99]`

Data Types: `single` | `double`

## Output Arguments

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

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### Student’s t cdf

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 cdf of the 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 falls in the interval [–∞, x].

## Alternative Functionality

• `tcdf` is a function specific to the Student's t distribution. Statistics and Machine Learning Toolbox™ also offers the generic function `cdf`, which supports various probability distributions. To use `cdf`, specify the probability distribution name and its parameters. Note that the distribution-specific function `tcdf` is faster than the generic function `cdf`.

• Use the Probability Distribution Function app to create an interactive plot of the cumulative distribution function (cdf) or probability density function (pdf) for a probability distribution.

## Version History

Introduced before R2006a