Student’s t distribution

William Gosset published this under the pseudonym “Student” in 1908 to model small-sample uncertainty. It looks like a Normal but has heavier tails — more room for extreme values when you don’t yet know the population variance.

f (x; ν) = \frac{Γ ( \frac{ν + 1}{2} )}{ν π Γ ( \frac{ν}{2} )} (1 + \frac{x ^{2}}{ν})^{- (ν + 1) /2}

t_3.0 Normal(0, 1)

ν (degrees of freedom) 3.0 n (samples) 1000

Bars: 1000 samples via inverse CDF. Solid: Student's t PDF. Dashed: Standard Normal for reference. As

ν

grows, the two curves merge.

What to notice

Small $ν$ means fat tails. At $ν = 1$ it’s the Cauchy distribution — no finite mean. At $ν = 2$ it has a mean but infinite variance.
Large $ν$ means Gaussian. The two curves are visually indistinguishable once $ν ≳ 30$ . That’s why textbook z-tests and t-tests converge when n is large.
Algebraic tails. The density falls off like $∣ x ∣^{- (ν + 1)}$ — a polynomial decay, in contrast to the Normal’s exponential one.

Where it comes from

If you take a standard Normal $Z$ and divide by the square root of an independent chi-squared scaled by its degrees of freedom, you get a t:

T = \frac{Z}{V / ν}, Z \sim N (0, 1), V \sim χ_{ν}^{2}

Plug in the sample mean and sample variance of n iid Normal draws and you get the classic t-statistic. The degrees of freedom $ν$ = n − 1 capture how much the uncertainty about $σ$ inflates the distribution of the mean.