Finding the covariance and correlation of two random variables

Question

Let $X$ be a random variable that has a standard uniform distribution $U(0,1)$, let $Y = X^k$, $k > 0$. I have performed the random variable transformation receiving $g(y) = \frac{1}{k}y^{\frac{1}{k}-1}$. I have also managed to calculate $\sigma_X = \frac{1}{2\sqrt{3}}$, $\sigma_Y = \frac{k}{\sqrt{2k+1}(k+1)}$. I know that $cov(X,Y) = \mathsf E[XY] - \mathsf E[X]\mathsf E[Y]$. How can I find the $E(XY)$?

Answer 1

You just have to remember what it is $Y$.

$$ Cov(X,Y) = Cov(X,X^k)= E(XX^k) - E(X)E(X^k) =E(X^{k+1}) - E(X)E(X^k) $$ and apply what they suggest in comments