Question

The joint distribution of X and Y is given by x/y 1 3 1 0.06 0.42 0.12 2 0.04 0.28 0.08 1. Are X and Y independent or dependent? 2. Prove your answer in part 1.

Answer 1

Solution Ke say Xey are independent if P(19) = Plt. PW for all xiy where P(X, Y) P(x) ply) a joint distribution of Xin marginal distribution of t. marginal distribution of y 2 PX) 0.60 0.06 0.042 0.12 - 0.04 0.28 0 08 0.40 2 0.10 ply) 020 0.70 7 > for pair (11) Play = 0.06 PCX). ply) = P(1) P(1) = 0.60 X 10.10 = 0.06 Ply, 4) = P P14 = 0.06 for pair (1,2) p(x,y)= 0.42 PEN 0.60 X 0.70 = 0.42 Plx, y) = PIX). Ply) = 0.42 pals (13) Plx, y = 0.12 POPY) = 0.60 X 0.20 = 0.12 P(x, y) = PIX). Ply) 0.12.

> for pair (2, 1) Pixi4)=0.04 P(x) ply)0.40 X 10.10 = 0.04 P(x, y) = P(x), ply)= 0.04. > for pair P(2, 2) P(XIYO = 0.28 PU) Ply) = 0.40X0.70 = 0.28 P(x} PLY)= 0.28 > for pair (2,3) Pixi4 = 0.08 P(x).DIY)= 0.40 0.20 p(x,y) =P). Ply) 20.08 0.08 from O to Pixi4) - PIX) ply) X & Y are independent all