# Proving Convarience of a Dice Throw

• Oct 26th 2007, 05:12 AM
Proving Convarience of a Dice Throw
Ok...

I have this question on a tutorial that i cant really figure out...

A fair die is thrown twice, and let X and Y be the resulting numbers. Let U = X - Y and V = X + Y. Show that Cov(U,V) = 0. Are U and V independant?

Now i know that if they are independant then this proves Cov(U,V) = 0 but i cant figure out if they are, and if they're not then ive no idea how to prove Cov(U,V) = 0.

Can anyone help on this?

Cheers
• Oct 26th 2007, 12:40 PM
CaptainBlack
Quote:

Ok...

I have this question on a tutorial that i cant really figure out...

A fair die is thrown twice, and let X and Y be the resulting numbers. Let U = X - Y and V = X + Y. Show that Cov(U,V) = 0. Are U and V independant?

Now i know that if they are independant then this proves Cov(U,V) = 0 but i cant figure out if they are, and if they're not then ive no idea how to prove Cov(U,V) = 0.

Can anyone help on this?

Cheers

$\displaystyle Cov(U,V) = E((U-\bar{U})(V-\bar{V}))=E((X-Y)(X+Y-7))$

.......... $\displaystyle =E(X^2-Y^2-7X+7Y)=E(X^2)-E(Y^2)-7E(X)+7E(Y)$

but $\displaystyle E(X^2)=E(Y^2),\ E(X)=E(Y)$, so:

$\displaystyle Cov(U,V)=0$

(and they are not independent)

RonL