Continuous random vector

Question 1

Continuous random vector
▪ Definition

Answer 1

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

Question 2

Continuous random vector
▪ Properties

Answer 2

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

Question 3

Continuous random vector
▪ Expected value

Answer 3

If g: Rⁿ –> R

E[g(x₁, x₂, …, xₙ)] =
∫g(x₁, x₂, …, xₙ)*f(x₁, x₂, …, xₙ) dx₁,dx₂,…,dxₙ

Question 4

Continuous random vector
▪ Independent

Answer 4

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 )