Documentation

# tpdf

Student's t probability density function

## Syntax

```y = tpdf(x,nu) ```

## Description

`y = tpdf(x,nu)` returns the probability density function (pdf) 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 have the same size. A scalar input is expanded to a constant array with the same dimensions as the other inputs.

## Examples

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The mode of the Student's t distribution is at x = 0. This example shows that the value of the function at the mode is an increasing function of the degrees of freedom.

`tpdf(0,1:6)`
```ans = 1×6 0.3183 0.3536 0.3676 0.3750 0.3796 0.3827 ```

The t distribution converges to the standard normal distribution as the degrees of freedom approaches infinity. How good is the approximation for $\nu$ equal to 30?

`difference = tpdf(-2.5:2.5,30)-normpdf(-2.5:2.5)`
```difference = 1×6 0.0035 -0.0006 -0.0042 -0.0042 -0.0006 0.0035 ```

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### Student’s t pdf

The probability density function (pdf) of the Student's t distribution is

`$y=f\left(x|\nu \right)=\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu }{2}\right)}\frac{1}{\sqrt{\nu \pi }}\frac{1}{{\left(1+\frac{{x}^{2}}{\nu }\right)}^{\frac{\nu +1}{2}}}$`

where ν is the degrees of freedom and Γ( · ) is the Gamma function. The result y is the probability of observing a particular value of x from a Student’s t distribution with ν degrees of freedom.