Var[X] is defined as:
Similarly, Var[X + Y] is defined as:
Expanding:
\[E[{\color{maroon} X^2} + 2{\color{maroon} X}{\color{darkblue} Y} + {\color{darkblue} Y^2}] - E[{\color{maroon} X}+{\color{darkblue} Y}]E[{\color{maroon} X}+{\color{darkblue} Y}] \]
Using the linearity of expectation:
\[\begin{gather} E[{\color{maroon} X^2}] + 2E[{\color{maroon} X}{\color{darkblue} Y}] + E[{\color{darkblue} Y^2}] - (E[{\color{maroon} X}]+E[{\color{darkblue} Y}])(E[{\color{maroon} X}]+E[{\color{darkblue} Y}]) \\ E[{\color{maroon} X^2}] + 2E[{\color{maroon} X}{\color{darkblue} Y}] + E[{\color{darkblue} Y^2}] - (E[{\color{maroon} X}]^2 + 2E[{\color{maroon} X}]E[{\color{darkblue} Y}] + E[{\color{darkblue} Y}]^2)\\ E[{\color{maroon} X^2}] + 2E[{\color{maroon} X}{\color{darkblue} Y}] + E[{\color{darkblue} Y^2}] - E[{\color{maroon} X}]^2 - 2E[{\color{maroon} X}]E[{\color{darkblue} Y}] - E[{\color{darkblue} Y}]^2 \\ (E[{\color{maroon} X^2}] - E[{\color{maroon} X}]^2) + (E[{\color{darkblue} Y^2}] - E[{\color{darkblue} Y}]^2) + (2E[{\color{maroon} X}{\color{darkblue} Y}] - 2E[{\color{maroon} X}]E[{\color{darkblue} Y}]) \end{gather} \]
Since X and Y are independent:
\[\begin{gather} (E[{\color{maroon} X^2}] - E[{\color{maroon} X}]^2) + (E[{\color{darkblue} Y^2}] - E[{\color{darkblue} Y}]^2) + (2E[{\color{maroon} X}]E[{\color{darkblue} Y}] - 2E[{\color{maroon} X}]E[{\color{darkblue} Y}]) \\ (E[{\color{maroon} X^2}] - E[{\color{maroon} X}]^2) + (E[{\color{darkblue} Y^2}] - E[{\color{darkblue} Y}]^2) \end{gather} \]
Since Var[X] = E[X2]-E[X]2:
