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t distribution

t Distribution

Probability Density FunctionThe formula for the probability density function of the t distribution isf(x) = (1 + x**2/nu)**(-(nu+1)/2)/[B(0.5,0.5*nu)*SQRT(nu)]
where B is the beta function and nu is a positive integer shape parameter. The formula for the beta function is
B(alpha,beta) = INTEGRAL(t**(alpha-1)*(1-t)**(beta-1)dt) where the integration is from 0 to 1
In a testing context, the t distribution is treated as a "standardized distribution" (i.e., no location or scale parameters). However, in a distributional modeling context (as with other probability distributions), the t distribution itself can be transformed with alocation parameter, mu, and a scale parameter, sigma.
The following is the plot of the t probability density function for 4 different values of the shape parameter.
plot of the t probability density function for 4 different values of the shape parameter
These plots all have a similar shape. The difference is in the heaviness of the tails. In fact, the t distribution with nu equal to 1 is aCauchy distribution. The t distribution approaches a normaldistribution as nu becomes large. The approximation is quite good for values of nu > 30.
Cumulative Distribution FunctionThe formula for the cumulative distribution function of the tdistribution is complicated and is not included here. It is given in theEvans, Hastings, and Peacock book.The following are the plots of the t cumulative distribution function with the same values of nu as the pdf plots above.
plot of the t cumulative distribution function with the same values of nu as the pdf plots above
Percent Point FunctionThe formula for the percent point function of the t distribution does not exist in a simple closed form. It is computed numerically.The following are the plots of the t percent point function with the same values of nu as the pdf plots above.
plot of the t percent point function with the same values of nu as the pdf plots above
Other Probability FunctionsSince the t distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit the formulas and plots for the hazard, cumulative hazard, survival, and inverse survival probability functions.
Common Statistics
Mean0 (It is undefined for nu equal to 1.)
Median0
Mode0
RangeInfinity in both directions.
Standard DeviationSQRT(nu/(nu-2))It is undefined for nu equal to 1 or 2.
Coefficient of VariationUndefined
Skewness0. It is undefined for nu less than or equal to 3. However, the t distribution is symmetric in all cases.
Kurtosis3*(nu-2)/(nu-4)It is undefined for nu less than or equal to 4.
Parameter EstimationSince the t distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit any discussion of parameter estimation.
CommentsThe t distribution is used in many cases for the critical regions for hypothesis tests and in determining confidence intervals. The most common example is testing if data are consistent with the assumed process mean.
SoftwareMost general purpose statistical software programs support at least some of the probability functions for the t distribution.

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