Random Sample

Definition

A realization of i.i.d. Random Vector

Properties

Let $\mathbf{x}=(X_1,X_2,\dots ,X_m{)}^{⊺},\mathbf{y}=(Y_1,Y_2,\dots ,Y_n{)}^{⊺}$ be random samples, $A$ be $k\times m$ constant matrix, and $B$ be $p\times n$ constant matrix, then the Expected Value and Covariance Matrix of random vectors become

• $E(A\mathbf{x})=AE(\mathbf{x})$
• $\mathrm{Var}(A\mathbf{x})=A\mathrm{Cov}(\mathbf{x})A^{⊺}$
• $\mathrm{Cov}(A\mathbf{x},B\mathbf{y})=A\mathrm{Cov}(\mathbf{x},\mathbf{y})B^{⊺}$