Let X and Y have a joint density given by f(X,Y)=a(1−X), 0≤X≤1,0≤Y ≤1

and zero elsewhere. Where, k is a constant.

(a) Find the value of a that makes this a probability density function. Use this

value for a in the pdf to work out the remaining questions.

(b) Find the marginal distribution of X and Y, i.e. f(X) and f(Y ).

(c) Find E(X), E(Y ), E(Y |X) and V ar(X|Y ).

(d) Find E[6X −12Y] and Var(2X −3Y)

(e) Find Cov(XY ) and the correlation ρxy

(f) Are X and Y independent? Show this using two alternative ways.

For part (a) I'm getting a value of a=-2 but I don't think that it's correct. I haven't moved any further in the question as all the others are dependent on this answer. Any help would be great!