# tpdf

Student's t probability density function

## 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