Chapter 4.4, Problem 18E

Let $f\left(x,y\right)=\frac{1}{8},0\le y\le 4,y\le x\le y+2,$ be the joint pdf of X and Y.

(a) Sketch the region for which $f\left(x,y\right)>0$.

(b) Find $f_X\left(x\right)$, the marginal pdf of X.

(c) Find $f_Y\left(y\right)$, the marginal pdf of Y.

(d) Determine $h\left(y|x\right)$

,the conditional pdf of Y, given that $X=x$

(c) Determine $g\left(x|y\right)$, the conditional pdf of X, given that $Y=y$.

(f) Compute $y=E\left(Y|x\right)$, the conditional mean of Y, given that $X=x$.

(g) Compute $x=E\left(X|y\right)$, the conditional mean of X, given that $Y=y$.

(h) Graph $y=E\left(Y|x\right)$ on your sketch in part (a). Is $y=E\left(Y|x\right)$ linear?

(I) Graph $x=E\left(X|y\right)$ on your sketch in part (a). Is $x=E\left(X|y\right)$ linear?