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 = {Xi} with joint pdf or pmf f(X) is independent if and only if there exist functions fi(xi) such that f(X) = Π fi(xi). In such cases, fi(xi) 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 MX(t) and MY(t), then the moment generating function for Z = X+Y is MZ(t) = MX(t)MY(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 Xi are independent normal random variables with means μi and variances σi2, then X=Σ(aiXi+bi) is a normal random variable with mean Σ(aiμi+bi) and variance Σ(aiσi2).
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