Confusing title, sorry I know.
Essentially given 3 random variables $X$, $Y$, and $Z$, where $X$ and $Y$ are independent, and $Z = XY$, what's the covariance of $(X, Z)$ (or $(X, XY)$)?
I know the basic properties like Cov$(X,X) =$ Var$(X)$ and how Cov$(X,Y) = 0$ since X and Y are independent. But I'm having trouble putting the two together (if that's even possible). Any insight into this relationship is appreciated. Thank you!
Just use the definition of covariance: $cov(X,Y) = \mathop{\mathbb{E}}(XY)-\mathop{\mathbb{E}}(X)\mathop{\mathbb{E}}(Y)$. In your case, it should be $cov(X,Z) = \mathop{\mathbb{E}}(XZ) - \mathop{\mathbb{E}}(X)\mathop{\mathbb{E}}(Z) = \mathop{\mathbb{E}}(X^2Y) - \mathop{\mathbb{E}}(X)\mathop{\mathbb{E}}(X)\mathop{\mathbb{E}}(Y) = (\mathop{\mathbb{E}}(X^2))\mathop{\mathbb{E}}(Y)-(\mathop{\mathbb{E}}(X))^2\mathop{\mathbb{E}}(Y) = var(X)\mathop{\mathbb{E}}(Y)$ To see why the second last equality holds, i.e. the independence of $X$ and $Y$ implies independence of $X^2$ and $Y$, you can refer to these posts: (1) If $X$ and $Y$ are independent then $f(X)$ and $g(Y)$ are also independent. (2) If $X$ and $Y$ are independent. How about $X^2$ and $Y$? And how about $f(X)$ and $g(Y)$?. I hope this helps :)
