Question

Question 1 This question asks you to analyze a causal system and derive the covariances/correlations be- tween the variables in the system. In Figure 1, the arrows represent directions of causality. For example, Y is a function E, and X1. These functions are given in the block of equations below. Please answer the following questions as a function of the parameters (01,02,1,3,7): a What are V[Y] and V[X2]? b What are CovX1,Y) and Cov[X1, X2]? c What is Cov[X2,Y)? d What are Cor[X1,Y) and Cor[X2,Y]? e Show that Cor[X1,Y] > Cor[X2, Y) for all values of the parameters E1, E2 and X1 are all independent of each other E1 ~ N(0,0) X1 Poisson() Y = BX1 + E1 E2 ~ N(0,0) X2 = 7X1 + E2 E1 X1 E2 Y X2 Figure 1: Question 1 Causal Diagram

Answer 1

ANSWER:-
E UN (0,07). X, UP(d) E U N (0,0) 2 Y=BX TE X2=8X1TER (a) V(4) V(Bxit E1) = 8 v Cx, JI VLE) [.XHE, are independent = 8², a to = 2 +BA V(+2) = V(X+1 + f2) = 8v (X) + V (E2) (xf Ez are independent tou Fother's (B) Cov. (H,Y) - cov(x1, BXiTE) Bu(+1)+cov (x1, E1) = Bato exit Ei are independent] cou H, tz) = coulx 18 x +€2) = 8U(X) + con (Xi, E2) =flto ("xif Ez are independent car

(e) cou(x2,Y) = cov (XXIHE , BX1+E) = BV V(Xi) + V cove xv, FD + B cov (X1, 2) + Cove Eife) =Brd trotißioto Exufi fr are independent of each other. 1 BV (d) cor (X1,4) cortil) ✓ Vexi) VLY) ſit or 718) Ito? XB2- cor (2,4) con tha, Y) TV (x2) VLY) ABV North Lor+132)

. สำนักการ AB2 top Ito 132 18² It oi Ara litou Ito? < :) xr² 182 AB? yotl Vue Iton dr² -DL ito? AB² ABL a) .com (X2;Y) & cor (X,Y) = cor (x,y) 2 cor (x2,Y)