Given that E[Y]=2 and Var [Y] =3, find E[(2Y + 1)^2]

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