Var[X] is defined as:
Similarly, Var[X + Y] is defined as:
Lets expand:
Since X + Y are independent:
Now lets cancel out 2[XY]:
Since Var[X] = E[X2]-E[X]2:
Proof That Var[X + Y] = Var[X] + Var[Y] Where X And Y Are Independent Variables
Var[X] is defined as:
Similarly, Var[X + Y] is defined as:
Lets expand:
Since X + Y are independent:
Now lets cancel out 2[XY]:
Since Var[X] = E[X2]-E[X]2: