... Rnare independent if and onlyif (2) FX1,···,Xm(x1, ,xm)=FX1(x1) ···FXm(xm) for all xi∈ Rn,i=1, ,m.If the random variables have densities, (2) is equivalent to(3) fX1,···,Xm(x1, ... (X):=Ω|X −E(X)|2dPthe variance of X.Observe thatV (X)=E(|X −E(X)|2)=E(|X|2) −|E(X)|2.LEMMA (Chebyshev’s inequality). If X is a random variable and 1 ≤ p<∞, thenP (|X|≥λ) ≤1λpE(|X|p) ... distribution, with mean m and covariance matrix C. We then writeX is an N(m, C) random variable.LEMMA. Let X :Ω→ Rnbe a random variable, and assume that its distribution func-tion F = FXhas...