# Adding variances

1. Jul 20, 2012

### waealu

Hi, I have a quick question about variance.

If you are trying to compute Var[3X-Y] where Var[X]=1 and Var[Y]=2,
the correct formula is 32*Var[X] + Var[Y],
but why do you add the variances of x and y?

If you were computing Var[3X+Y] would it also be 32*Var[X] + Var[Y] ?

Thanks, Erik

2. Jul 20, 2012

### Staff: Mentor

Your textbook probably has a theorem about Var[X + Y] and Var[kX]
Yes.

3. Jul 20, 2012

### chiro

Hey waealu.

For the proof, the easiest way is to use VAR[X] = E[X^2] - {E[X]}^2 and then prove the identity. You do it for the continuous and discrete cases and you'll get the same result for both.

Intuitively though, the variance is a squared positive quantity which means that the variance has to always increase (and recall that variances are always positive for a random variable).

4. Jul 21, 2012

### Ray Vickson

These are only true if X and Y are independent (or, at least, uncorrelated) random variables. For independent (or uncorrelated) X and Y and constants a, b we have
$\text{Var}[aX + bY] =a^2 \text{Var}(X) + b^2 \text{Var}(Y).$

RGV

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