I don't even know the subject of this question, possibly Bivariate Normal

• Nov 11th 2009, 12:12 PM
chella182
I don't even know the subject of this question, possibly Bivariate Normal
Let $\displaystyle X$ and $\displaystyle Y$ be random variables with zero means, variances $\displaystyle \sigma_{x}^{2}$, $\displaystyle \sigma_{y}^{2}$ and correlation $\displaystyle \rho$. Find the value of $\displaystyle a$ which minimises $\displaystyle Var(Y-aX)$ and evaluate this minimum value. Deduce the minimum value $\displaystyle Var(X-bY)$ and the value of $\displaystyle b$ at which the minimum is attained. Investigate the cases $\displaystyle \rho=0,\pm1$ explicitly. Comment generally.

Literally no idea on this :( thanks for any help in advance.
• Nov 11th 2009, 12:44 PM
Laurent
Quote:

Originally Posted by chella182
Let $\displaystyle X$ and $\displaystyle Y$ be random variables with zero means, variances $\displaystyle \sigma_{x}^{2}$, $\displaystyle \sigma_{y}^{2}$ and correlation $\displaystyle \rho$. Find the value of $\displaystyle a$ which minimises $\displaystyle Var(Y-aX)$ and evaluate this minimum value. Deduce the minimum value $\displaystyle Var(X-bY)$ and the value of $\displaystyle b$ at which the minimum is attained. Investigate the cases $\displaystyle \rho=0,\pm1$ explicitly. Comment generally.

Literally no idea on this :( thanks for any help in advance.

Why not explicitate $\displaystyle {\rm Var}(Y-aX)$? You have $\displaystyle {\rm Var}(X+Y)={\rm Var}(X)+2{\rm Cov}(X,Y)+{\rm Var}(Y)$ and $\displaystyle {\rm Var}(\lambda X)=\lambda^2 {\rm Var}(X)$, hence you should be able to write $\displaystyle {\rm Var}(Y-aX)$ as a polynomial of degree 2 in the variable $\displaystyle a$.
• Nov 11th 2009, 01:47 PM
chella182
Quote:

Originally Posted by Laurent
Why not explicitate $\displaystyle {\rm Var}(Y-aX)$? You have $\displaystyle {\rm Var}(X+Y)={\rm Var}(X)+2{\rm Cov}(X,Y)+{\rm Var}(Y)$ and $\displaystyle {\rm Var}(\lambda X)=\lambda^2 {\rm Var}(X)$, hence you should be able to write $\displaystyle {\rm Var}(Y-aX)$ as a polynomial of degree 2 in the variable $\displaystyle a$.

Alright, so I'm at $\displaystyle a^2Var(Y)-2aCov(X,Y)+Var(x)$. Obviously $\displaystyle Var(Y)=\sigma_{y}^2$ and $\displaystyle Var(X)=\sigma_{x}^2$, but is there another way to express the $\displaystyle Cov(X,Y)$? Like $\displaystyle \sigma_{x}\sigma_{y}\rho$ or something? And even after that am I differentiating or factorising or something? (Worried) I hate questions like this.
• Nov 11th 2009, 06:22 PM
BERRY
Quote:

Originally Posted by chella182
Alright, so I'm at $\displaystyle a^2Var(Y)-2aCov(X,Y)+Var(x)$. Obviously $\displaystyle Var(Y)=\sigma_{y}^2$ and $\displaystyle Var(X)=\sigma_{x}^2$, but is there another way to express the $\displaystyle Cov(X,Y)$? Like $\displaystyle \sigma_{x}\sigma_{y}\rho$ or something? And even after that am I differentiating or factorising or something? (Worried) I hate questions like this.

Since you want to find $\displaystyle a$ which minimizes $\displaystyle Var(Y-aX)$, you should take the first derivative of $\displaystyle Var(Y-aX)$ and set it to 0 and solve for a.
• Nov 11th 2009, 06:27 PM
theodds
Rewrite the covariance the way you suggested, take the derivative, and set her to zero. Bling Blang Blaow.
• Nov 12th 2009, 03:03 AM
chella182
Yeah, I got it in the end, cheers :)