Random vectors and Joint distribution

If X and Y are discrete random variables, also (X, Y) is discrete, and we denote ƿ(x,y) its density More explicitly p(x,y) = P(X=x, Y=y) called the joint density of X and Y ▪ In this context the densities ƿₓ and ƿy are called marginal densities

1) ▪ pₓ (x) = Σy pₓ,y (x,y) ▪ py (y) = Σₓ pₓ,y (x,y) 2) Let X and Y be two discrete random variables, taking values in E and F respectively, and ▪ f: E * F –> R then ▪ E(f(x,y)) = Σₓ,y f(x,y) pₓ,y (x,y)

Let: ▪ X be a random variable with values in E ▪ Y be a random variable with values in F X,Y are independent if for every ▪ A ⊆ E ▪ B ⊆ F P(XϵA, YϵB) = P(XϵA) * P(YϵB)

1) X and Y are independent iff ▪ ∀XϵA, ∀YϵB ▪ pₓ,y(x,y) = pₓ(x) * py(y) 2) Let X,Y ▪ independent ▪ discrete ▪ real valued random variables Then E(X,Y) = E(X) * E(Y)

Let X,Y be two real valued random variables The covariance between X and Y is defined by Cov(X,Y) = E[(X-E(X)(Y-E(Y))] | Cov(X,Y) = E(XY) - E(X)E(Y)

1) Cov(X,Y) = Cov(Y, X) 2) Cov(X,aY + bZ) = aCov(X,Y)+bCov(X,Z) 3) Cov(X,X) = Var(X) 4) If X and Y are independent the Cov(X,Y)=0 If Cov(X,Y) = 0 we say that X and Y are uncorrelated We have seen that independent –> uncorrelated (but the inverse is not necessarily true)

The correlation coefficient detects exact or approximate linear relations between random variables ⍴(X,Y) = ( Cov(X,Y) ) / ( √Var(X) * √Var(Y) )

▪ ⍴(aX + b, cY + d) = ⍴(X,Y) ▪ -1 ⩽ ⍴(X,Y) ⩽ 1 ▪ ⍴(X,Y) = 1 iff Y=aX + b for some a>0 ▪ ⍴(X,Y) = -1 iff Y=aX + b for some a<0

Theorem Var(X+Y)= Var(X) + Var(Y) + 2Cov(X,Y) Corollary Var(X+Y)= Var(X) + Var(Y) If X and Y are uncorrelated (in particular if they are independent)

▪ If X₁ ,X₂, …, Xₙ are discrete random variables, their joint density is defined by px₁ ,x₂, …, xₙ (x₁ ,x₂, …, xₙ) = P(X₁ = x₁ ,X₂ = x₂, …, Xₙ = xₙ) ▪ pxi = Σx₁,x₂,…,xₙ p(x₁ ,x₂, …, xₙ) ▪ E[f(x₁ ,x₂, …, xₙ)] = Σx₁,x₂,…,xₙ f(x₁ ,x₂, …, xₙ)*px₁,x₂,…,xₙ(x₁ ,x₂, …, xₙ) ▪ X₁ ,X₂, …, Xₙ are said to be independent for all choices of A₁ ,A₂, …, Aₙ P( X₁ϵ A₁, X₂ϵ A₂, …, Xₙϵ Aₙ) = ℿₖ P(Xₖ ϵ Aₖ) px₁ ,x₂, …, xₙ (x₁ ,x₂, …, xₙ) = ℿ pₓₖ (xₖ)