The Student’s t distribution is a family of curves depending on a single parameter ν (the degrees of freedom).
The Student’s t distribution uses the following parameter.
Parameter | Description |
---|---|
ν = 1, 2, 3,... | Degrees of freedom |
The probability density function (pdf) of the Student's t distribution is
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.
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
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
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
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 ν equal to 1. The Cauchy distribution has an undefined mean and variance.
The cumulative distribution function (cdf) of Student’s t distribution is
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].
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
Use tinv
to compute the
inverse cdf of the Student’s t distribution.
p = .95; nu = 50; x = tinv(p,nu)
x = 1.6759
The mean of the Student’s t distribution is
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
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
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')
random
| tcdf
| tinv
| tpdf
| trnd
| tstat