Let ((X,Y)) be a continuous random vector with joint pdf [ f_{X,Y}(x,y) = k x y {for } 0 x 1, 0 le y le 1, ] and zero otherwise. (a) Determine the constant (k) that makes (f_{X,Y}(x,y)) a valid pdf. Verify the conditions for a valid pdf by integrating over the unit square. (b) Find the marginal pdfs (f_X(x)) and (f_Y(y)). Are (X) and (Y) independent? Justify your answer. (c) Compute the conditional pdf of (Y) given (X=x), denoted (f_{Y|X}(y|x)). Use this to calculate (E[Y|X=x]) and then determine (E[Y]) by applying the law of iterated expectation. (d) Define a new random variable (Z = X + Y). Derive the pdf of (Z) by convolution or geometric reasoning. Sketch the pdf and indicate its support