This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English version of the page.

Note: This page has been translated by MathWorks. Click here to see
To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

Student's t Distribution

Overview

The Student’s t distribution is a family of curves depending on a single parameter ν (the degrees of freedom).

Parameters

The Student’s t distribution uses the following parameter.

ParameterDescription
ν = 1, 2, 3,...Degrees of freedom

Probability Density Function

Definition

The probability density function (pdf) of the Student's t distribution is

y=f(x|ν)=Γ(ν+12)Γ(ν2)1νπ1(1+x2ν)ν+12

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 a Student’s t distribution with ν degrees of freedom.

Plot

This plot shows how changing the value of the degrees of freedom parameter ν alters the shape of the pdf. Use tpdf to compute the pdf for values x equals 0 through 10, for three different values of ν. Then plot all three pdfs on the same figure for a visual comparison.

x = [0:.1:10];
y1 = tpdf(x,5);   % For nu = 5
y2 = tpdf(x,25);  % For nu = 25
y3 = tpdf(x,50);  % For nu = 50

figure;
plot(x,y1,'Color','black','LineStyle','-')
hold on
plot(x,y2,'Color','red','LineStyle','-.')
plot(x,y3,'Color','blue','LineStyle','--')
legend({'nu = 5','nu = 25','nu = 50'})
hold off

Random Number Generation

Use trnd to generate random numbers from the Student’s t distribution. For example, the following generates a random number from a Student’s t distribution with degrees of freedom ν equal to 10.

nu = 10;
r = trnd(nu)
r = 1.0585

Relationship to Other Distributions

As the degrees of freedom ν goes to infinity, the t distribution approaches the standard normal distribution.

If x is a random sample of size n from a normal distribution with mean μ, then the statistic

t=x¯μs/n

where x¯ is the sample mean and s is the sample standard deviation, has Student's t distribution with n – 1 degrees of freedom.

The Cauchy distribution is a Student’s t distribution with degrees of freedom ν equal to 1. The Cauchy distribution has an undefined mean and variance.

Cumulative Distribution Function

Definition

The cumulative distribution function (cdf) of Student’s t distribution is

p=F(x|ν)=xΓ(ν+12)Γ(ν2)1νπ1(1+t2ν)ν+12dt

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 will fall in the interval [–∞, x].

Plot

This plot shows how changing the value of the parameter ν alters the shape of the cdf. Use tcdf to compute the cdf for values x equals 0 through 10, for three different values of ν. Then plot all three cdfs on the same figure for a visual comparison.

x = [0:.1:10];
y1 = tcdf(x,5);   % For nu = 5
y2 = tcdf(x,25);  % For nu = 25
y3 = tcdf(x,50);  % For nu = 50

figure;
plot(x,y1,'Color','black','LineStyle','-')
hold on
plot(x,y2,'Color','red','LineStyle','-.')
plot(x,y3,'Color','blue','LineStyle','--')
legend({'nu = 5','nu = 25','nu = 50'})
hold off

Inverse cdf

Use tinv to compute the inverse cdf of the Student’s t distribution.

p = .95;
nu = 50;
x = tinv(p,nu)
x = 1.6759

Mean and Variance

The mean of the Student’s t distribution is

mean=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

var=νν2

for degrees of freedom ν greater than 2. If ν is less than or equal to 2, then the variance is undefined.

Use tstat to compute the mean and variance of a Student’s t distribution. For example, the following computes the mean and variance of a Student’s t distribution with degrees of freedom ν equal to 10.

nu = 10;
[m,v] = tstat(nu)
m = 0
v = 1.2500

Example

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 goes to 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 = 25. Compute the pdf for a standard normal distribution.

x = -5:0.1:5;
y1 = tpdf(x,5);
y2 = tpdf(x,15);
z = normpdf(x,0,1);

Plot the Student's t pdfs and the standard normal pdf on the same figure. The standard normal pdf has shorter tails than the Student's t pdfs.

plot(x,y1,'-.',x,y2,'--',x,z,'-')
legend('Student''s t Distribution with \nu=5', ...
    'Student''s t Distribution with \nu=25', ...
    'Standard Normal Distribution','Location','best')
title('Student''s t and Standard Normal pdfs')

See Also

| | | | |

Related Examples

More About