Continuous random vector

Suppose we have a random vector X = (X₁, X₂, …, Xₙ) where X₁, X₂, …, Xₙ are real-valued random variables We say that X is a continuous random vector if there exists a function fₓ(x₁, x₂, …, xₙ) such that for any A ⊆ Rⁿ P(XϵA) = ₐ∫ fₓ(x₁, x₂, …, xₙ) dx₁, dx₂, …, dxₙ

If (x,Y) is a 2-dimensional continuous random vector with joint density fₓ,y Then ▪ both X and Y are continuous random variables ▪ and fₓ(x) = -∞∫∞ fₓ,y (x,y) dy ▪ and fy(y) = -∞∫∞ fₓ,y (x,y) dx ▪ The joint density is not determined by the marginal densities ▪ It may happen that X and Y are continuous random variables, but (X,Y) is not continuous random vector

If g: Rⁿ –> R E[g(x₁, x₂, …, xₙ)] = ∫g(x₁, x₂, …, xₙ)*f(x₁, x₂, …, xₙ) dx₁,dx₂,…,dxₙ

The components X₁, X₂, …, Xₙ of a continuous random vectors are independent iff fx₁, x₂, …, xₙ(x₁, x₂, …, xₙ) = ℿₖ fₖ(xₖ) If X and Y are continuous and independent, then E(XY) = E(X) * E(Y) ( –> Cov(X,Y) = 0 )