Consider two continuous random variablesX,Ywhich have the uniform distribution on quadrilateral with vertices (0,0),(0,1),(1,2),(1,1).

LetZ=X+Y.

a) By conditioning onX(i.e.Z|X), use the law of total variance to find the variance ofZ=X+Y(in particular, you need to derive and justifyfZ|Xand whatever else is used along the computations).

b) FindFZandfZ.

c) Evaluate the following three quantities:

E(E(Z|X)) (E(Z|X) should have been computed in parta)).

E(Z) =E(X+Y) =E(X) +E(Y) where each of these two expectations should be

computed deriving and usingfXandfY(i.e. the PDFs forXandY).