Consider the model illustrated in Fig. 1.9 on p. 13.

Let's refer to the paths coming from

Using the structural equations style presented on p. 23, we have

andA=tT+uU

(B=tT+vV.

Recalling that all of the variables have mean 0 and variance 1, and noting that

in agreement with the result given on p. 13. (ρ_{AB}= Cov(A,B) = E(AB) = E([tT+uU][tT+vV]) =t^{2}E(T^{2}) =t^{2}Var(T) =t^{2},

We also have that (noting that

1 = Var(and soA) = Var(tT+uU) =t^{2}Var(T) +u^{2}Var(U) =t^{2}+u^{2},

Now consider the model illustrated in Fig. 1.8 on p. 11.

Using the structural equations style presented on p. 23, we have

C=aA+bB+dX.

Recalling that all of the variables have mean 0 and variance 1 (and noting that

equalsρ_{AC}= Cov(A,C) = E(AC)

E(in agreement with the result given on p. 11. (A[aA+bB+dX]) =aE(A^{2}) +bE(AB) +dE(AX) =aVar(A) +bρ_{AB}+dρ_{A,X}=a+bc.

Noting that

Var(and soaA+bB+dX) =a^{2}Var(A) +b^{2}Var(B) +d^{2}Var(X) + 2abCov(A,B) =a^{2}+b^{2}+d^{2}+ 2abc,

in agreement with p. 12.d= ( 1 - (0.4)^{2}- (0.5)^{2}- 2(0.4)(0.5)(0.5) )^{1/2}= 0.39^{1/2},