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

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

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

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

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

Literally no idea on this thanks for any help in advance.
Why not explicitate ${\rm Var}(Y-aX)$? You have ${\rm Var}(X+Y)={\rm Var}(X)+2{\rm Cov}(X,Y)+{\rm Var}(Y)$ and ${\rm Var}(\lambda X)=\lambda^2 {\rm Var}(X)$, hence you should be able to write ${\rm Var}(Y-aX)$ as a polynomial of degree 2 in the variable $a$.

3. Originally Posted by Laurent
Why not explicitate ${\rm Var}(Y-aX)$? You have ${\rm Var}(X+Y)={\rm Var}(X)+2{\rm Cov}(X,Y)+{\rm Var}(Y)$ and ${\rm Var}(\lambda X)=\lambda^2 {\rm Var}(X)$, hence you should be able to write ${\rm Var}(Y-aX)$ as a polynomial of degree 2 in the variable $a$.
Alright, so I'm at $a^2Var(Y)-2aCov(X,Y)+Var(x)$. Obviously $Var(Y)=\sigma_{y}^2$ and $Var(X)=\sigma_{x}^2$, but is there another way to express the $Cov(X,Y)$? Like $\sigma_{x}\sigma_{y}\rho$ or something? And even after that am I differentiating or factorising or something? I hate questions like this.

4. Originally Posted by chella182
Alright, so I'm at $a^2Var(Y)-2aCov(X,Y)+Var(x)$. Obviously $Var(Y)=\sigma_{y}^2$ and $Var(X)=\sigma_{x}^2$, but is there another way to express the $Cov(X,Y)$? Like $\sigma_{x}\sigma_{y}\rho$ or something? And even after that am I differentiating or factorising or something? I hate questions like this.
Since you want to find $a$ which minimizes $Var(Y-aX)$, you should take the first derivative of $Var(Y-aX)$ and set it to 0 and solve for a.

5. Rewrite the covariance the way you suggested, take the derivative, and set her to zero. Bling Blang Blaow.

6. Yeah, I got it in the end, cheers