A normal distribution, but not exactly a normal distribution

Overview

  • Topic: A distribution like the normal but not exactly: the Student's t-distribution.

Key Concepts

  • Symmetric about 0 and centered at 0.
  • Shape depends on degrees of freedom (
    ν\nu).
  • Heavier tails than the normal for finite ν\nu.
  • Converges to the normal as ν→∞\nu \to \infty.

Formulas

  • Probability density function:
    f(t∣ν)=Γ(ν+12)νπ Γ(ν2)(1+t2ν)−ν+12f(t|\nu) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\,\Gamma\left(\frac{\nu}{2}\right)}\left(1 + \frac{t^2}{\nu}\right)^{-\frac{\nu+1}{2}}
  • Mean:
    E[T] = 0\quad (\nu > 1)
  • Variance:
    \operatorname{Var}(T) = \frac{\nu}{\nu-2}\quad (\nu > 2)
  • Relation to normal:
    Tν→ν→∞N(0,1)T_{\nu} \xrightarrow{\nu \to \infty} N(0,1)

Use Cases

  • Unknown population standard deviation with small sample sizes.
  • Used in t-tests and small-sample confidence intervals.

Quick Take

  • Heavier tails for small ν\nu; approaches normal as ν\nu grows.