3. The following equations are incorrect since the r.h.s. is unfinished. Please add whatever is appropriate to the r.h.s. so that the equations hold. Assume that (X, Y) have joint p.d.f. fx,y (s, t). (a) 1 = fx,y(s, t) (b ) Fx, y (u, v) = fx, x (s, t) (c) fx (s) = fx, x (s, t) (d) E[X] = fx(s) (e) Elg (X, Y) ] = fx,x (s, t)(f ) fy ( t) = fx, x (s, t) (g) fxly=+ (s) = fx, x(s, t) (h) E[X | Y = t] = fxly=(s) (i) E[X2 | Y = 1 ] = fxly=(s) (i) E[E[X | Y] ] = E[X | Y = +] (In the above, you should be thinking of E[X | Y] as a function of Y and applying LOTUS. BTW, E[E[X | Y] ] = E[X]. )