Joint p.d.f. and Independent Random Variables

♣ Let X and Y be two discrete r.v.'s and let R be the corresponding space of X and Y. The joint p.d.f. of X = x and Y = y, denoted by f (x, y) = P (X = x, Y = y), has the following properties: (a) 0 ≤ f (x, y) ≤ 1, f (x, y) ≥ 0 for − ∞ < x, y < ∞. ∞ −∞ ∞ −∞ f (x, y)dxdy = 1. y f (x, y) (∞ −∞ f (x, y)dy), x ∈ R x. ♣ The marginal p.d.f. of Y is defined as f Y (y) = x f (x, y) (∞ −∞ f (x, y)dx), y ∈ R y. ♣ The random variables X and Y are independent iff f (x, y) ≡ f X (x)f Y (y) for x ∈ R x , y ∈ R y .