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
Best Answer
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?