Question

(a) Suppose that X, Y and Z are random variables whose joint distribution is continuous with density fxyz. Write down appropriate definitions of of (i) fxyz, density of the joint distribution of X and Y given Z, and (ii) fxyz, density of the distribution of X given both Y and Z. Assuming the expectations exist, prove the tower property: E[E[X|Y, 2]|2] = E[X|2], by expressing both sides using the densities you have defined. Suppose that X and Y are independent given Z. What form does the joint density fxyz take in this case? Verify that in this case, E[X|Y, Z] = E[X|Z].

Answer 1

$f_{XY/Z}=f_{X/YZ}\times f_{Y/Z}$

$\therefore f_{Y/Z}=\frac{f_{XY/Z}}{f_{Y/Z}}$

$E[E[X/Y,Z]/Z]=\sum E[X/Y=y,Z=z]P[Y=y/Z=z]=\sum xE[X/Y=y,Z=z]P[Y=y/Z=z]=\sum x\frac{P(X=x,Y=y,Z=z)}{P(Y=y,Z=z)}\frac{P(Y=y,Z=z)}{P(Z=z)}=\sum x\frac{P(X=x,Y=y,Z=z)}{P(Z=z)}=\sum x\frac{P(X=x,Z=z)}{P(Z=z)}=\sum xP[X=x/Z=z]=E[X/Z]$

When X and Y are independent of z,then f_x/z = f_x and f_y/z = fy