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## Student's t Distribution

### Definition

Student's t pdf is

where Γ( · ) is the Gamma function.

### Background

The t distribution is a family of curves depending on a single parameter ν (the degrees of freedom). As ν goes to infinity, the t distribution approaches the standard normal distribution.

W. S. Gossett discovered the distribution through his work at the Guinness brewery. At the time, Guinness did not allow its staff to publish, so Gossett used the pseudonym "Student."

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.

### Example

The plot compares the t distribution with ν = 5 (solid line) to the shorter tailed, standard normal distribution (dashed line).

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