Student’s t Distribution — $ t_\nu $
The Student’s t distribution arises as a ratio of a Normal variable and the square root of an independent Chi-square variable. It is the fundamental distribution for mean inference with unknown variance and a canonical example of a pivot.
Definition
Degrees of freedom:
- $ \nu \in \mathbb{N} $
Support:
\(x \in \mathbb{R}\)
PDF:
\(f(x)
=
\frac{\Gamma\!\left(\frac{\nu+1}{2}\right)}
{\sqrt{\nu\pi}\,\Gamma\!\left(\frac{\nu}{2}\right)}
\left(1+\frac{x^2}{\nu}\right)^{-(\nu+1)/2}\)
Construction from Normal and Chi-square
Let \(Z \sim \mathcal{N}(0,1), \quad U \sim \chi^2_\nu, \quad Z \perp U.\)
Then \(T = \frac{Z}{\sqrt{U/\nu}} \sim t_\nu\)
This representation is the definition used in proofs.
Relationship to Other Distributions
Connection to Chi-square
- Denominator involves $ \chi^2_\nu $
- Degrees of freedom come from variance estimation
Connection to Normal
\[t_\nu \Rightarrow \mathcal{N}(0,1) \quad \text{as } \nu \to \infty\]Heavier tails for small $ \nu $.
Moments
- Mean: $ 0 $ for $ \nu > 1 $
- Variance: \(\operatorname{Var}(T) = \frac{\nu}{\nu-2}, \quad \nu>2\)
Higher moments exist only for sufficiently large $ \nu $.
Symmetry
\[T \overset{d}{=} -T\]All odd moments (when defined) are zero.
Pivotal Quantity (Key Use)
If \(X_1,\dots,X_n \stackrel{iid}{\sim} \mathcal{N}(\mu,\sigma^2),\) then \(\frac{\bar{X}-\mu}{S/\sqrt{n}} \sim t_{n-1}\)
This is a pivot, free of unknown parameters.
Role in Inference
Used for:
- Confidence intervals for $ \mu $
- Hypothesis testing when variance is unknown
- Regression coefficient inference
Likelihood Perspective
The t distribution arises from integrating out the variance in a Normal–Inverse-Gamma model.
Interpretation:
- t is a scale mixture of Normals
- Explains heavy tails
Tail Behavior
\[\mathbb{P}(|T|>x) \sim C x^{-\nu} \quad \text{as } x \to \infty\]Heavier tails than Normal.
Multivariate Extension (Recognition Only)
- Multivariate t arises from Normal with random scale
- Used in robust modeling
(No formulas required for prelims.)
Key Theorems and Facts (Prelim-Relevant)
- Ratio of Normal and Chi-square
- Pivot for mean with unknown variance
- Converges to Normal
- Heavy tails
- Scale-mixture interpretation
Exam Takeaways
- Unknown variance ⇒ t
- Degrees of freedom = sample size − 1
- Think “Normal divided by Chi-square”
- Heavier tails protect against variance uncertainty
- Central object for classical inference
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