Given that E[Y]=2 and Var [Y] =3, find E[(2Y + 1)^2] Multiply through E[(2Y + 1)^2] = E[4y^2 + 4y + 1] We can take the expected value of each term E[4y^2] + E[4y] + E[1] For the first term, we have: 4E[Y^2] We define the Var[Y] = E[Y^2] - (E[Y])^2 Rearrange this term, we have E[Y^2] = Var[Y] + (E[Y])^2 E[Y^2] = 3+ 2^2 E[Y^2] = 3+ 4 E[Y^2] = 7 So our first term is 4(7) = 28 For the second term using expected value rules of separating out a constant, we have 4E[Y] = 4(2) = 8 For the third term, we have: E[1] = 1 Adding up our three terms, we have: E[4y^2] + E[4y] + E[1] = 28 + 8 + 1 E[4y^2] + E[4y] + E[1] = 37