\[E[x+y] = \int \int(x+y)p(x,y)dxdy \]
\[= \int \int (x+y)p(x)(y)dxdy\]
\[= \int \int xp(y)dy p(x)dx + \int \int yp(x)p(y)dxdy\]
\[= \int(\int p(y)dy)xp(x)dx + \int (\int p(x)dx)yp(y)dy\]
\[= \int xp(x)dx + \int yp(y)dy\]
\[= E[x] + E[y]\]
\newline
\[var[x + y] = \int \int (x + y - E[x + y])^{2}p(x,y)dxdy\]
\[=\int \int((x + y)^{2} - 2(x+y)E[x+y] + E^{2}[x + y])p(x,y)dxdy \]
\[=\int \int(x + y)^{2}p(x,y)dxdy -2E[x +y]\int\int (x + y)p(x,y)dxdy +E^{2}[x+y] \]
\[=\int \int (x + y)^{2}p(x,y)dxdy -E^{2}[x + y]\]
\[=\int \int (x^{2} + 2xy + y^{2})p(x)p(y)dxdy - E^{2}[x + y]\]
\[=\int (\int p(y)dy)x^{2}p(x)dx + \int\int 2xyp(x)p(y)dxdy + \int(\int p(x)dx)y^{2}p(y)dy - E^{2}[x + y]\]
\[= E[x^{2}] + E[y^{2}] + E^{2}[x + y]\int \int 2xyp(x)p(y)dxdy\]
\[= E[x^{2}] + E[y^{2}] - (E[x]+E[y])^{2} + \int \int 2xyp(x)p(y)dxdy\]
\[=E[x^{2}] -E^{2}[x] + E[y^{2}] - E^{2}[y] -2E[x]E[y] + 2\int \int xyp(x)p(y)dxdy\]
\[=var[x] + var [y] -2E[x]E[y] + 2(\int xp(x)dx)(\int yp(y)dy)\]
\[=var[x] + var [y]\]