An important thing to understand is the following set of statements: “if random variables x and y are independent, they are uncorrelated. if random variables x and y are uncorrelated and jointly gaussian, they are independent.” this problem gently walks you through the nuances of this thinking. (a) often, a counterexample to an intuitive rule is some pathological case. however, this example occurs in standard communication systems; your cell phone is generating these random variables right now. let θ be a continuous random variable uniformly distributed on [0, 2π]. let x = cos θ and y = sin θ. show that, for this x and y , x and y are uncorrelated but not independent. (hint: as part of the solution, you will need to find e[x], e[y ] and e[xy ]. this should be pretty easy; if you find yourself trying to find fx(x) or fy (y), you are doing this the (very) hard way.) (b) (note: this part is ridiculously easy and i know that you saw it in lecture, but i give it to you so that you will hopefully remember it.) recall that two random variables x a