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A random vector X ~ N(μ, Σ) has characteristic function φ(t) = exp(it'μ - ½t'Σt). The MVN is closed under linear transformations and marginals. A joint distribution is MVN if every linear combination of components is univariate normal. The MVN is fundamental in statistical inference because the sample mean vector is MVN for large samples.
You know the univariate normal N(μ, σ²): a bell-shaped distribution centered at μ with spread controlled by σ². The multivariate normal distribution (MVN) extends this to random vectors X = (X₁, ..., Xₙ)'. The cleanest definition: X is MVN if every linear combination a'X = a₁X₁ + ... + aₙXₙ is univariate normal for any fixed vector a. This says the MVN is "normal in every direction" — no matter how you project the joint distribution onto a line, you get a normal curve.
The MVN is parameterized by a mean vector μ ∈ ℝⁿ (where the distribution is centered) and a covariance matrix Σ ∈ ℝⁿˣⁿ (which must be positive semidefinite). The diagonal entries are variances: Σᵢᵢ = Var(Xᵢ). The off-diagonal entries capture correlations: Σᵢⱼ = Cov(Xᵢ, Xⱼ). When Σ is diagonal, the components are independent normals. A positive Σᵢⱼ means Xᵢ and Xⱼ tend to move together; negative means they move in opposite directions.
From your joint distributions work, you know that marginals are obtained by integrating out other variables — often a painful computation. For the MVN, marginals are trivial: if X ~ N(μ, Σ) and you split X into subvectors X = (X₁, X₂)', then X₁ ~ N(μ₁, Σ₁₁) where μ₁ is the corresponding subvector of μ and Σ₁₁ is the corresponding submatrix of Σ. You just read off the relevant pieces. No integration required. This is a major computational advantage of the MVN.
The closure under linear transformations (from your linear transformations prerequisite) is equally powerful: if X ~ N(μ, Σ) and A is a matrix, then AX ~ N(Aμ, AΣA'). This single fact explains why the sample mean X̄ = (1/n)1'X is normal when the data are iid normal — it is a linear transformation of the data vector. More generally, any quantity computed as a linear function of normally distributed data inherits normality. The characteristic function φ(t) = exp(it'μ − ½t'Σt) encodes the entire distribution and makes this closure trivial to prove: φ_{AX}(t) = φ_X(A't), and substituting confirms the form. It also shows the MVN is completely determined by its first two moments — mean and covariance — since all higher cumulants vanish.