Two i.i.d Random Variables : Is covariance between a function (applied to one R.V) with the the R.V. the same for both Random Variables?

Question

Let $X,Y$, both $$\Omega \to \mathbb{R}^k$$ be i.i.d random variables. Let $f$ be a function from $\mathbb{R}^k \to \mathbb{R}$.

Is $\operatorname{cov}(f(X), X) = \operatorname{cov}(f(Y),Y)$?

If we use definition $$ \operatorname{cov}(X,Y) = E\left [(X-E[X])(Y-E[Y])\right ] = E[XY]-E[X]E[Y]$$ then I want to say yes, because both $X,Y$ have the same expectation.

Thanks.

Answer 1

$$\operatorname{cov}(f(X), X) = E[Xf(X)] - E[X]E[f(X)]$$

$$\operatorname{cov}(f(Y), Y) = E[Yf(Y)] - E[Y]E[f(Y)]$$

$Xf(X)$ and $Yf(Y)$ are iid.

$f(X)$ and $f(Y)$ are iid.

Assuming all the expectations are defined, we have our desired result.

I don't think independence is needed, but I think it is if you instead asked if

$$\operatorname{cov}(f(X), Y) = \operatorname{cov}(f(Y), X)$$