Expectation of of four random variables

Question

Four random variables $X_1, X_2, Y_1$, $Y_2$, and they are all independent to each other. I'd like to know the expectation of $(X_1X_2+Y_1Y_2)^2$, i.e. $E[(X_1X_2+Y_1Y_2)^2]$. I have two expressions for this expectation and I don't know which one is correct or both are wrong.

First, $\begin{align}E[(X_1X_2+Y_1Y_2)^2] ~=~& \iiiint (x_1x_2+y_1y_2)^2 f_{X_1X_2Y_1Y_2}(x_1x_2y_1y_2)dx_1dx_2dy_1dy_2 \\[1ex]~=~& \iiiint (x_1x_2+y_1y_2)^2 f_{X_1}(x_1)f_{X_2}(x_2)f_{Y_1}(y_1)f_{Y_2}(y_2)dx_1dx_2dy_1dy_2\end{align}$.

Second, $\begin{align}E[(X_1X_2+Y_1Y_2)^2] ~=~& E[X_1^2X_2^2]+E[Y_1^2Y_2^2]+2E[X_1X_2Y_1Y_2] \\[2ex]~=~&{ \iint\ x_1^2x_2^2 f_{X_1}(x_1)f_{X_2}(x_2)dx_1dx_2 + \iint\ y_1^2y_2^2 f_{Y_1}(y_1)f_{Y_2}(y_2)dx_1dx_2\\+2\iiiint x_1x_2y_1y_2 f_{X_1}(x_1)f_{X_2}(x_2)f_{Y_1}(y_1)f_{Y_2}(y_2)dx_1dx_2dy_1dy_2}\end{align}$

Answer 1

They are equivalent, in as much as $$\int_{\boldsymbol \Omega} (x_1 x_2 + y_1 y_2)^2 f_{\boldsymbol X}(\boldsymbol x) \, d \boldsymbol x = \int_{\boldsymbol \Omega} (x_1 x_2)^2 f_{\boldsymbol X}(\boldsymbol x) \, d \boldsymbol x + \int_{\boldsymbol \Omega} (y_1 y_2)^2 f_{\boldsymbol X}(\boldsymbol x) \, d \boldsymbol x + 2\int_{\boldsymbol \Omega} (x_1 x_2 y_1 y_2) f_{\boldsymbol X}(\boldsymbol x) \, d \boldsymbol x.$$ More simply put, recall that $$\int f(x) + g(x) \, dx = \int f(x) \, dx + \int g(x) \, dx,$$ whenever the RHS integrals exist.

Answer 2

Just expand to use the Linearity of Expectation, and then employ the fact of independence.

$$\begin{align}&\mathsf E((X_1X_2+Y_1Y_2)^2) \\[1ex] =~& \mathsf E(X_1^2X_2^2+2X_1X_2Y_1Y_2+Y_1^2Y_2^2)\\[1ex] =~& \mathsf E(X_1^2)\,\mathsf E(X_2^2)+2\,\mathsf E(X_1)\,\mathsf E(X_2)\,\mathsf E(Y_1)\,\mathsf E(Y_2)+\mathsf E(Y_1^2)\,\mathsf E(Y_2^2)\end{align}$$

So $$\begin{align}&\iiiint (x_1x_2+y_1y_1)^2 f_{X_1}(x_1)f_{X_2}(x_2)f_{Y_1}(y_1)f_{Y_2}(y_2)\mathsf d y_2\mathsf d y_1\mathsf d x_2\mathsf d x_1 \\[1ex] =~& {\int x_1^2f_{X_1}(x_1)\mathsf d x_1{\cdot}\int x_2^2f_{X_2}(x_2)\mathsf d x_2 \\+ 2\int x_1f_{X_1}(x_1)\mathsf d x_1{\cdot}\int x_2f_{X_2}(x_2)\mathsf d x_2{\cdot}\int y_1f_{Y_1}(y_1)\mathsf d y_1{\cdot}\int y_2f_{Y_2}(y_2)\mathsf d y_2 \\+ \int y_1^2f_{Y_1}(y_1)\mathsf d y_1{\cdot}\int y_2^2f_{Y_2}(y_2)\mathsf d y_2}\end{align}$$