If X and Y are random variables with correlation coefficient 0.7, each of which has variance 6, what is the variance of X−Y? Enter your answer as a decimal.
Using the information given, I was able to determine the Covariance of X and Y to be 4.2
I thought maybe the variance of X-Y would be 0 but that's too easy.
Any suggestions, I feel I'm close and I understand the formulas
On
Are you familiar with inproducts and norms? Then think of covariance as inproduct, standarddeviation as norm and variation as square of the norm.
$$\text{Var}\left(X-Y\right)=\text{Cov}\left(X-Y,X-Y\right)=\text{Cov}\left(X,X\right)-2\text{Cov}\left(X,Y\right)+\text{Cov}\left(Y,Y\right)=$$ $$\text{Var}X-2\sigma_{X}\sigma_{Y}\text{ Correl}\left(X,Y\right)+\text{Var}Y$$
Analogously:
$$\left\Vert u-v\right\Vert ^{2}=\left(u-v,u-v\right)=\left(u,u\right)-2\left(u,v\right)+\left(v,v\right)=$$ $$\left\Vert u\right\Vert ^{2}-\left\Vert u\right\Vert \left\Vert v\right\Vert \left(\frac{u}{\left\Vert u\right\Vert },\frac{v}{\left\Vert v\right\Vert }\right)+\left\Vert v\right\Vert ^{2}$$
Hint:
Write out the variance as much as you can, then look for quantities with known values. We start from $$ \newcommand{\Var}{\text{Var}}\newcommand{\E}{\mathbb{E}}\newcommand{\Cov}{\text{Cov}} \Var[X-Y]=\E[(X-Y)^2]-(\E[X-Y])^2. $$ Now, we can multiply these out and use linearity of the expectation to get: $$ \Var[X-Y]=\E[X^2]-2\E[XY]+\E[Y^2]-(\E[X])^2+2\E[X]\E[Y]-(\E[Y])^2. $$ Now, we know the values for $\Var[X]=\E[X^2]-(\E[X])^2$, $\Var[Y]=\E[Y^2]-(\E[Y])^2$, and $\Cov[X,Y]=\E[XY]-\E[X]\E[Y]$. Can you see how to rewrite $\Var[X-Y]$ in terms of those functions?