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

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

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 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

$$t=\frac{\overline{x}-\mu}{s/\sqrt{n}}$$

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 *ν*equal to 1. The
Cauchy distribution has an undefined mean and variance.

The cumulative distribution function (cdf) of 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 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

$$\text{mean}=0$$

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

$$\mathrm{var}=\frac{\nu}{\nu -2}$$

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

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

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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