STA107 Lectures 12 and 13 Joint Distributions & Independence

STA107 Lectures 12 and 13 Joint Distributions & Independence Joint pmf pxy(x,y) = P(X=x, Y=y) Example Joint CDF Fxy(x,y) = P(X ≤ x, Y ≤ y)

Properties of the joint pmf • 0 ≤ pxy(x,y) ≤ 1 Axiom 1 • sl;kj’ Axioms 2 & 3 Properties of the joint CDF • Limit as k1 ∞ & k2 ∞ of Fxy(k1,k2) = 1 • Limit as k1 -∞ & k2 -∞ of Fxy(k1, k2) = 0 • Limit as k1 ∞ of Fxy(k1,y) = Fy(y) • If a<b & c<d then Fxy(a,c) ≤ Fxy(b,d)

Marginal Distributions px(x) = Sy pxy(x, y) add down columns py(y) = Sx pxy(x, y) add across rows Example

Conditional Distributions Recall – events A and B, P(A|B) = P(AB)/P(B) Let A be the event in S corresponding to X = x Let B be the event in S corresponding to Y = y • P(A|B) = P(X=x|Y=y) = P(X=x, Y=y)/P(Y=y) = pxy(x,y)/ py(y) • px|y(x|y) = pxy(x,y)/ py(y) Note: Y is a constant, X is a random variable Also - py|x(y|x) = pxy(x,y)/ px(x)

Example

Independent Random Variables Recall – events A and B independent  P(AB) = P(A)P(B) Let A be the event in S corresponding to X = x Let B be the event in S corresponding to Y = y Random variables X and Y independent  P(X=x,Y=y) = P(X=x)P(Y=y)  pxy(x,y) = px(x) py(y) Note: this must hold for all x and for all y

Example pxy(2,1) = 0 px(2)py(1) = (0.25)(0.15) = 0.0375 ≠ 0 X and Y are dependent random variables

Example Example

STA107 Lectures 12 and 13 Joint Distributions & Independence