Can var(x+y) Be Less Than or Equal to 2(var(x) + var(y))?

[kbilsback5]

Hi, I was hoping that someone might be able to please help me with this proof.

Prove that var(x+y) ≤ 2(var(x) + var(y)).

So far I have:

var(x+y) = var(x) + var(y) + 2cov(x,y)

where the cov(x,y) = E(xy) - E(x)E(y), but I'm not really sure to go from there.
Any insight would be very helpful!

Thanks!

 

[noowutah]

Let $\alpha=var(x),\beta=var(y),\gamma=cov(x,y)$. According to the Cauchy-Schwarz inequality (see http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality for the proof), $\gamma^{2}\leq{}\alpha\beta$. We want to show $\alpha{}+\beta{}+2\gamma\leq{}2(\alpha{}+\beta)$, which follows directly from

$2\gamma\leq{}2(\gamma^{2})^{\frac{1}{2}}\leq{}2(\alpha\beta)^{\frac{1}{2}}\leq{}2(\frac{\alpha{}+\beta}{2})\leq{}\alpha{}+\beta$

where $(\alpha\beta)^{\frac{1}{2}}\leq{}\frac{\alpha{}+\beta}{2}$ follows from the well-known fact that the geometric mean is always smaller than the arithmetic mean (see http://www.cut-the-knot.org/pythagoras/corollary.shtml for proof).

 

[kbilsback5]

Awesome! Thanks so much for your help!

 

[noowutah]

You are welcome.