For the last line of this answer, how do we know that that is true? [Why does -->Which is not even dependent on v. Hence, they are independent.] holds true?

3. (10 points) Suppose that the pdf of Y, conditional on X = r, is N(x, r-). Suppose further that the marginal distribution of X is Unif (0, 1). (a) Use conditioning to find E(Y), var(Y) and cov(X, Y).(b) U = g(X, Y) = Y/X, V = h(X, Y) = X T = V. y = UV J(x, y) = ah = OHI = T 3 fu.v(u, v) = fxx(x, y)| - x] 1 1 - expl - 272 ( y - x)2} 1 exp{ - 1 202 (40 - v)2} 1 exp{ - 1 V/2AT = exp{u - u"/2 - 1/2} Which is not even dependent on v. Hence, they are independent.(b) The joint pdf of X and Y can be derived to be fxx(x, y) = exp 2:12 " - 2 ) 2 ) , VER, OC Let U = Y/X and V = X. Derive the joint pdf of U and V and use it to decide if U and V are independent