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 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 νequal 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

Compute the pdf for a Student's t distribution with parameter nu = 5, and for a standard normal distribution.

x = -5:0.1:5;
y = tpdf(x,5);
z = normpdf(x,0,1);

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

figure;
plot(x,y,'-',x,z,'-.')

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