Given two random variables $X$ and $Y$ I need to prove $$E[(X+Y)(X+Y)^T] = \int \int (x+y)(x+y)^Tp_{X,Y}(x,y)dxdy$$ where $p_A(.)$ is the probability density function of the random variable $A$.
My method :
Let $U = X + Y$
$E[(X+Y)(X+Y)^T] = E[UU^T] = \int uu^tp_U(u)du$
$p_U(u) = \int p_{X,Y}(x,u-x)dx$, therefore
$E[UU^T] = \int uu^tp_U(u)du = \int [uu^T\int p_{X,Y}(x,u-x)dx]du$
I'm stuck here and don't know how to proceed.
Result taken from Equation 17 of https://www.researchgate.net/profile/Puneet_Singla3/publication/224639311_A_quasi-Gaussian_Kalman_filter/links/02e7e52000e0d06c78000000/A-quasi-Gaussian-Kalman-filter.pdf
In general, for any continuous function $f(x,y)$ and two random variables $X,Y$ with joint density $p_{X,Y}(x,y)$, $$ E \left[ f(X,Y) \right] = \iint f(x,y) p_{X,Y}(x,y)\; dx \; dy $$ This is just a multivariate version of the Law of the Unconscious Statistician.