Question

1. Consider a discrete bivariate random variable (X,Y) with the joint pmf given by the table: Y X 1 2 4 1 0 0.1 0.05 2 0.2 0.05 0 4 0.1 0 0.05 8 0.3 0.15 0 Table 0.1: p(, y) a) Find marginal distributions of X and Y, p(x) and pay respectively. b) Find the covariance and the correlation between X and Y.

Answer 1

- 1 1 0 0.1 o.os x 2 4 2 0.2 0.05 0 0.25 4 0.1 0 0.05 0.15 - wo 8 IP (K) 0.3 0.6 0.rs 0.3 o 0.45 II ply) 10.15 (a) The marginal distributions X and Y are calculated as and ply) respectively in above table. of bek) the (b) cov(x, y) = (x4) – E(X) Ely) E(XY) - 1x1x 0 + 1x2 x 0-2 + 1 x 4 x o.1 + 1x 8x0.3 0 + 2 x 8 x 0.15 + 2x 1 xool + 2x 2 x o.ost 2 x 4 x + 4x1xo.o5 + 4 x 2 x 0 + 4 x 4 x 0.05 + 4 x 8 x 0 - 0 + 0.4 & 0.4 & 2.4 + 0.2 + 0.2 to & 2.4 t 0.2 + 0 + 0.8 & o E ( XY) - 7

+ 4 X 0 E(X) = Exp (R) 2 1 x 0.6 + 2x0.3 - 0.6 + 0.6 + E (X) = 1.6 0.4 E(X) = Erp Ox) = 12 x 0.6 & 22 x 0.3 = (x 0.6 + 4 x 0.3 + 42 x 0.1 + 16x 0.1 & 1.2 + 1.6 = = 0.6 3.4 vex) = E(X2) - [E(X)]? - 3.4 – (106) 2 = 3.4 – 2.56 11 1.14 – Jux) = √114 = 1.06 Ely) & Eyply) - x ools & 2 x 0.25 + 4x o.is + 8 x 0.45 - t o .s t 0.6 + 3.6 0.15 4.85 &

E (42) = 2y ²ply) = 12x 0.15 & 2² x + 8 2 x das 0.25 + 42 x 0.15 = 1o.ls + 4 x 0.25 +16 xo.15 + 64 x 0.43 = o.13 + 1 + 2-4 t 28.8 = 32.35 Vly) = (y2) – [E(4) ² = 32.35 – (4.85) 2 = 32.35 - 23.52 Ty- Suey) – 8.83 - 2.97 we have, Elx 4) = 7 E(X) = 1.6 Ely) - 4.85 covariance between X and 4 is cov (x, 4) - Elxy) - ECX) E (4) - T- s.6 x 4.85 - 7 - 7.16 given by

CON (x, y) = 0.76 correlation given x between by and y is 4 L x, y) = CON (x, y) X. Ty he have Love x, y) = -0.76 Ty = 2.97 1(x, y) = – 0.76 1.06 x 2.97 = -0.16 314 M(x, y) = - 0.24