Miscellaneous multivariate independence results

No named results here; these results could certainly be used without reference in any paper as they are foundational.

Multivariate independence: A set of random variables X = {X_i} with joint pdf or pmf f(X) is independent if and only if there exist functions f_i(x_i) such that f(X) = Π f_i(x_i). In such cases, f_i(x_i) are the marginal pdfs or pmfs for the elements of X.

Functions of independent random variables: If X and Y are independent random variables, then g(X) and h(Y) are independent.

Products of independent random variables: If X and Y are independent random variables, then E(g(X)h(Y)) = E(g(X))E(h(Y)).

Moments of independent random variables: If X and Y are independent random variables having moment generating functions M_X(t) and M_Y(t), then the moment generating function for Z = X+Y is M_Z(t) = M_X(t)M_Y(t).

That last one suggests a very important specific result, that the sum of two independent normal random variables is a normal random variable. The actual result can be stated stronger and extended to any linear combination of independent normals: If X_i are independent normal random variables with means μ_i and variances σ_i², then X=Σ(a_iX_i+b_i) is a normal random variable with mean Σ(a_iμ_i+b_i) and variance Σ(a_iσ_i²).