Calculates the excess kurtosis of the Student's t-Distribution.

## Syntax

**TDIST_XKURT**(

**v**)

- v
- is the degrees of freedom of the Student's t-Distribution (v > 4).

## Remarks

- TDIST_XKURT is declared as deprecated. Please, use DIST_XKURT as TDIST_XKURT is listed here only for backward compatibility.
- The probability density function of the Student's t-Distribution is defined as:

$$f(t) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu} \right)^{-(\nu+1)/2} $$

Where:

- $\Gamma (.)$ is the gamma function.
- $\nu $ is the degrees of freedom (i.e. shape parameter).

- The excess kurtosis of t-Distribution is defined as:

$$\gamma_2= \frac{6}{\nu-4}$$

Where:

- $\nu$ is the degrees of freedom.

**IMPORTANT**The Student's t-Distribution kurtosis is only defined for degrees of freedom values greater than 4.- Special Cases:
- $ \nu\to 4^+$

$$\lim_{\nu\to 4^+}\gamma_2(\nu)=+\infty$$ - $ \nu\to \infty $

$$\lim_{\nu\to +\infty}\gamma_2(\nu)=0$$

- $ \nu\to 4^+$

## Examples

**Example 1: **

##### Student's t-Distribution X-Kurtosis Plot

**Example 2: **

Formula | Description (Result) |
---|---|

=TDIST_XKURT(5) | Excess kurtosis with 5 degrees of freedom (6.000) |

=TDIST_XKURT(100) | Student t-dist approaches Normality as v >> 1 (0.063) |

=TDIST_XKURT(4.002) | Excess kurtosis increases as v approaches 4 (3000.000) |

## Files Examples

## Related Links

## References

- K.L. Lange, R.J.A. Little and J.M.G. Taylor. "Robust Statistical Modeling Using the t Distribution." Journal of the American Statistical Association 84, 881-896, 1989
- Hurst, Simon, The Characteristic Function of the Student-t Distribution, Financial Mathematics Research Report No. FMRR006-95, Statistics Research Report No. SRR044-95

## Comments

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