So I have this pdf, $f(x)=3x^2$ for $x\in (0,1)$ and I need to find $Cov(2X+7, X^2+3X-12)$. My main concern about how I answer this is, what is the joint pdf for these two distributions? I guess it's $f(x)$ but I wanted to confirm this.
But as for actually calculating the covariance, I first use the fact that constant terms can vanish, so that I instead calculate $Cov(2X, X^2+3X)$. I'll use the definition of covariance which tells me to compute
so I need to know each mean for $2X$ and $X^2+3X$. I know how to calculate those, so to avoid writing too much, let's assume I've found their values. Then I believe the integral I want to compute is
Is this correct?