Student's t Distribution

Overview

The Student’s t distribution is a one-parameter family of curves. This distribution is typically used to test a hypothesis regarding the population mean when the population standard deviation is unknown.

Statistics and Machine Learning Toolbox™ offers multiple ways to work with the Student’s t distribution.

• Use distribution-specific functions (tcdf, tinv, tpdf, trnd, tstat) with specified distribution parameters. The distribution-specific functions can accept parameters of multiple Student’s t distributions.

• Use generic distribution functions (cdf, icdf, pdf, random) with a specified distribution name ('T') and parameters.

Parameters

The Student’s t distribution uses the following parameter.

Parameter	Description	Support
nu (ν)	Degrees of freedom	ν = 1, 2, 3,...

Probability Density Function

The pdf of the Student's t distribution is

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

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.

For an example, see Compute and Plot Student's t Distribution pdf.

Cumulative Distribution Function

The cdf of the Student’s t distribution is

$$p=F(x|\nu )={\displaystyle {\int }_{-\infty }^x\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu }{2}\right)}\frac{1}{\sqrt{\nu \pi }}\frac{1}{{\left(1+\frac{t^2}{\nu }\right)}^{\frac{\nu +1}{2}}}dt},$$

where ν is the degrees of freedom and Γ( · ) is the Gamma function. The result p is the probability that a single observation from the t distribution with ν degrees of freedom falls in the interval [–∞, x].

For an example, see Compute and Plot Student's t Distribution cdf.

Inverse Cumulative Distribution Function

The t inverse function is defined in terms of the Student's t cdf as

$$x=F^{-1}(p|\nu )=\{x:F(x|\nu )=p\},$$

where

$$p=F(x|\nu )={\displaystyle {\int }_{-\infty }^x\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu }{2}\right)}\frac{1}{\sqrt{\nu \pi }}\frac{1}{{\left(1+\frac{t^2}{\nu }\right)}^{\frac{\nu +1}{2}}}dt},$$

ν is the degrees of freedom, and Γ( · ) is the Gamma function. The result x is the solution of the integral equation where you supply the probability p.

For an example, see Compute Student's t icdf.

Descriptive Statistics

The mean of the Student’s t distribution is μ = 0 for degrees of freedom ν greater than 1. If ν equals 1, then the mean is undefined.

The variance of the Student’s t distribution is $$\frac{\nu }{\nu -2}$$ for degrees of freedom ν greater than 2. If ν is less than or equal to 2, then the variance is undefined.

Examples

Compute and Plot Student's t Distribution pdf

Compute the pdf of a Student's t distribution with degrees of freedom equal to 5, 10, and 50.

x = [-5:.1:5];
y1 = tpdf(x,5);
y2 = tpdf(x,10);
y3 = tpdf(x,50);

Plot the pdf for all three choices nu on the same axis.

figure;
plot(x,y1,'Color','black','LineStyle','-')
hold on
plot(x,y2,'Color','red','LineStyle','-.')
plot(x,y3,'Color','blue','LineStyle','--')
xlabel('Observation')
ylabel('Probability Density')
legend({'nu = 5','nu = 10','nu = 50'})
hold off

Compute and Plot Student's t Distribution cdf

Compute the cdf of a Student's t distribution with degrees of freedom equal to 5, 10, and 50.

x = [-5:.1:5];
y1 = tcdf(x,5);
y2 = tcdf(x,10);
y3 = tcdf(x,50);

Plot the cdf for all three choices of nu on the same axis.

figure;
plot(x,y1,'Color','black','LineStyle','-')
hold on
plot(x,y2,'Color','red','LineStyle','-.')
plot(x,y3,'Color','blue','LineStyle','--')
xlabel('Observation')
ylabel('Cumulative Probability')
legend({'nu = 5','nu = 10','nu = 50'})
hold off

Compute Student's t icdf

Find the 95th percentile of the Student's t distribution with 50 degrees of freedom.

p = .95;   
nu = 50;   
x = tinv(p,nu)

x = 1.6759

Compare Student's t and Normal Distribution pdfs

The Student’s t distribution is a family of curves depending on a single parameter ν (the degrees of freedom). As the degrees of freedom ν approach infinity, the t distribution approaches the standard normal distribution.

Compute the pdfs for the Student's t distribution with the parameter nu = 5 and the Student's t distribution with the parameter nu = 15.

x = [-5:0.1:5];
y1 = tpdf(x,5);
y2 = tpdf(x,15);

Compute the pdf for a standard normal distribution.

z = normpdf(x,0,1);

Plot the Student's t pdfs and the standard normal pdf on the same figure.

plot(x,y1,'-.',x,y2,'--',x,z,'-')
legend('Student''s t Distribution with \nu=5', ...
    'Student''s t Distribution with \nu=15', ...
    'Standard Normal Distribution','Location','best')
xlabel('Observation')
ylabel('Probability Density')
title('Student''s t and Standard Normal pdfs')

The standard normal pdf has shorter tails than the Student's t pdfs.

Related Distributions

• Beta Distribution — The beta distribution is a two-parameter continuous distribution that has parameters a (first shape parameter) and b (second shape parameter). If Y has a Student's t distribution with ν degrees of freedom, then $$X=\frac{1}{2}+\frac{1}{2}\frac{Y}{\sqrt{\nu +Y^2}}$$ has beta distribution with the shape parameters a = ν/2 and b = ν/2. This relationship is used to compute values of the t cdf and inverse functions, and to generate t distributed random numbers.

• Cauchy Distribution — The Cauchy distribution is a two-parameter continuous distribution with the parameters γ (scale) and δ (location). It is a special case of the Stable Distribution with the shape parameters α = 1 and β = 0. The standard Cauchy distribution (unit scale and location zero) is the Student’s t distribution with degrees of freedom ν equal to 1. The standard Cauchy distribution has an undefined mean and variance.

  For an example, see Generate Cauchy Random Numbers Using Student's t.

• Chi-Square Distribution — The chi-square distribution is a one-parameter continuous distribution that has the parameter ν (degrees of freedom). If Z has a standard normal distribution and χ² has a chi-square distribution with degrees of freedom ν, then $$\text{t = }\frac{Z}{\sqrt{{\chi }^2/\nu }}$$ has a Student's t distribution with degrees of freedom ν.

• Noncentral t Distribution — The noncentral t distribution is a two-parameter continuous distribution that generalizes the Student's t distribution and has the parameters ν (degrees of freedom) and δ (noncentrality). Setting δ = 0 yields the Student's t distribution.

• Normal Distribution — The normal distribution is a two-parameter continuous distribution with the parameters μ (mean) and σ (standard deviation).

  As the degrees of freedom ν approach infinity, the Student's t distribution approaches the standard normal distribution (zero mean and unit standard deviation).

  For an example, see Compare Student's t and Normal Distribution pdfs

  If x is a random sample of size n from a normal distribution with mean μ, then the statistic $$t=\frac{\overline{x}-\mu }{s/\sqrt{n}}$$, where $$\overline{x}$$ is the sample mean and s is the sample standard deviation, has a Student's t distribution with n —1 degrees of freedom.

  For an example, see Compute Student's t Distribution cdf.

• t Location-Scale Distribution — The t location-scale distribution is a three-parameter continuous distribution with the parameters μ (mean), σ (scale), and ν (shape). If x has a t location-scale distribution with the parameters µ, σ, and ν, then $$\frac{x-\mu }{\sigma }$$ has a Student's t distribution with ν degrees of freedom.

References

[1] Abramowitz, Milton, and Irene A. Stegun, eds. Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. 9. Dover print.; [Nachdr. der Ausg. von 1972]. Dover Books on Mathematics. New York, NY: Dover Publ, 2013.

[2] Devroye, Luc. Non-Uniform Random Variate Generation. New York, NY: Springer New York, 1986. https://doi.org/10.1007/978-1-4613-8643-8

[3] Evans, Merran, Nicholas Hastings, and Brian Peacock. Statistical Distributions. 2nd ed. New York: J. Wiley, 1993.

[4] Kreyszig, Erwin. Introductory Mathematical Statistics: Principles and Methods. New York: Wiley, 1970.