All books

Professional

Apps About Coach Log in Start reading

Mathematics · Glossary

What is Gaussian vector?

Definition 23.10 University Mathematics — Year 3 · Chapter 23 — Characteristic Functions and the Central Limit Theorem

A random vector X=(X1,…,Xd)X = (X_1, \dots, X_d) is Gaussian if every linear combination ⟨t,X⟩=∑tiXi\langle t, X\rangle = \sum t_iX_i is a (possibly degenerate) real Gaussian variable. Its law is determined by the mean vector m=(EXi)m = (\E X_i) and the covariance matrix Σ=(Cov⁡(Xi,Xj))\Sigma = \bigl(\operatorname{Cov} (X_i, X_j)\bigr): indeed the characteristic function of the vector, φX(t)=Eei⟨t,X⟩\varphi_X(t) = \E\eu^{\iu\langle t, X\rangle}, is the value at 11 of the cf of ⟨t,X⟩\langle t, X\rangle:

φX(t)=exp⁡(i⟨t,m⟩−12 tTΣ t),\varphi_X(t) = \exp\Bigl(\iu\langle t, m\rangle - \tfrac12\,t^{\mathsf T}\Sigma\,t\Bigr),

and dd-dimensional characteristic functions are injective (same smoothing proof as Theorem 23.3, coordinatewise Gaussians).

Examples

Example 23.9 (Confidence intervals, honestly derived)

Poll nn independent voters; p^n=Sn/n\hat p_n = S_n/n estimates the true pp, with σ2=p(1−p)≤14\sigma^2 = p(1-p) \leq \frac14. The CLT gives, for large nn,

P(∣p^n−p∣≤z2n)  ≥  P(∣Sn−npσn∣≤z)⟶Φ(z)−Φ(−z),\P\Bigl(\abs{\hat p_n - p} \leq \frac{z}{2\sqrt n}\Bigr) \;\geq\; \P\Bigl(\Bigl|\frac{S_n - np}{\sigma\sqrt n}\Bigr| \leq z\Bigr) \longrightarrow \Phi(z) - \Phi(-z),

where Φ\Phi is the standard Gaussian distribution function. With z=1.96z = 1.96: asymptotic confidence 95%95\%, and margin 1.962n≤3%\frac{1.96}{2\sqrt n} \leq 3\% requires n≥(1.960.06)2≈1068n \geq \bigl(\frac{1.96}{0.06}\bigr)^2 \approx 1068 — the number behind every “±3\pm3 points, 95%95\%” one reads; compare Chebyshev’s 55565556 (Exercise 22.7). The n\sqrt n is universal: to halve the error, quadruple the sample — the same law that fixes Monte Carlo’s cost (Exercise 23.7).

Read in context →

Related terms