# tpdf

Student's t probability density function

## Syntax

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

## Description

example

````y = tpdf(x,nu)` returns the probability density function (pdf) of the Student's t distribution with `nu` degrees of freedom, evaluated at the values in `x`.```

## Examples

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The value of the pdf at the mode is an increasing function of the degrees of freedom.

The mode of the Student's t distribution is at x = 0. Compute the pdf at the mode for degrees of freedom `1` to `6`.

`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 approach infinity.

Compute the difference between the pdfs of the standard normal distribution and the Student's t distribution pdf with `30` degrees of freedom.

`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 ```

## Input Arguments

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

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

• To evaluate the pdfs 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, `tpdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `y` is the pdf 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 pdf at multiple values, specify `x` using an array.

• To evaluate the pdfs 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, `tpdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `y` is the pdf 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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pdf 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 `y` is the pdf 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 pdf

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

The 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 the Student’s t distribution with ν degrees of freedom.

## Alternative Functionality

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

• 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