# Multivariate Distributions

Let be an n-dimensional vector of random variables.
For all , the joint cumulative distribution function of X satisfies

Clearly it is straightforward to generalise the previous definition to join marginal distributions. For example, the join marginal distribution of and satisfies

If and is a partition of X then the conditional CDF of X_2 given X_1 satisfies
latex f_X (\bullet)latex X_2latex X_1latex f_{X_2 | X_1} (X_2 | X_1) = \frac{f_X (X)}{f_{X_1}(X_1)} = \frac{f_{X_1 | x_2}(X_1 | X_2) f_{X_2}(X_2)}{f_{X_1}(X_1)}latex F_{X_2 | X_1}(X_2 |X_1) = \int_{- \infty}^{x_{k+1}} \ldots \int_{- \infty}^{x_n} \frac{f_X (x_1, \ldots , x_k, u_{k+1}, \ldots , u_n)}{f_{X_1}(X_1)} du_{k+1} \dots du_nlatex f_{X_1}(\bullet)latex X_1latex f_{X_1} (x_1, \ldots , x_k) = \int_{- \infty}^{\infty} \ldots \int_{- \infty}^{\infty} f_X (x_1, \ldots , x_K u_{k+1}, \ldots , u_n) du_{k+1} \ldots du_nlatex F_X (x_1 \ldots , x_n) = F_{X_1} (x_1) \ldots F_{X_n}(x_n)latex f_X(\bullet)Latex f_X(x) = f_{X_1} (x_1) \ldots f_{X_n}(x_n)latex X_1latex X_2latex f_{x_2|x_1}(x_2 | x_1) = \frac{f_X (X)}{f_{X_1}(X_1)} = \frac{f_{X_1}(X_1) f_{X_2}(X_2)}{f_{X_1}(X_1)} = f_{X_2}(X_2)latex X_1latex X_2latex P(X \in A, Y \in B) = P(X \in A) P(Y \in B)latex P(X \in A, Y \in B)latex = \mathbb{E}[1_{X \in A} 1_{Y \in B}]latex = P(X \in A) P(Y \in B)latex X_1, \ldots, X_nlatex \mathbb{E}[f_1 (X_1) f_2(X_2) \ldots f_n(X_n)] = \mathbb{E}[f_1(X_1)] \mathbb{E}[f_2(X_2)] \ldots \mathbb{E}[f_n(X_n)]latex \mathbb{E}[f(X)g(Y)|Z] = \mathbb{E}[f(X)|Z]\mathbb{E}[g(Y)|Z]latex D_ilatex i^{th}latex D_ilatex P(D_1, \ldots, D_n |Z ) = P(D_1|Z) \ldots P(D_n|Z)Latex \mathbb{E}[X]:=(\mathbb{E}[X_1] \ldots \mathbb{E}[X_n])^Tlatex \sum := \mathrm{Cov}(X) := \mathbb{E}[(X- \mathbb{E}[X])(X – \mathbb{E}[X])^T]latex (i,j)^{th}latex \sumLatex X_ilatex X_jlatex \sum_{i, i} \ge 0latex x^T \sum x \ge 0latex x \in \mathbb{R}^nlatex \rho (X)latex (i, j)^{th}latex \rho_{ij} := \mathrm{Corr}(X_i, X_j)latex A \in \mathbb{R}^{k \times n}latex a \in \mathbb{R}^kLatex \mathbb{E}[AX +a] = A \mathbb{E} [X] + alatex \mathrm{Cov}(AX +a) = A \mathrm{Cov}(X) A^Tlatex \Rightarrow \mathrm{Var} (aX + bY) = a^2 \mathrm{Var}(X) + b^2 \mathrm{Var}(Y) + 2ab \mathrm{Cov}(X, Y)latex \mathrm{Cov}(X, Y)=0\$, and the converse is not true in general.