2.8 () Consider two variables x and y with joint distribution p(x, y). Prove the following two results E[x] = Ey [Ex[x|y]] (2.270)

var[x] = Ey [varx[x|y]] + vary [Ex[x|y]] . (2.271)

Here Ex[x|y] denotes the expectation of x under the conditional distribution p(x|y), with a similar notation for the conditional variance.