Let $X$ and $Y$ be two uniformly distributed random variables on $[0,1]$. Find $E(X^k)$ and $E(XY^k)$.

Question

Let $X$ and $Y$ be two uniformly distributed random variables on $[0,1]$. Find $E(X^k)$ and $E(XY^k)$.

How can you do this? Do you need the change of variables technique? I am a bit confused about that.

Answer 1

Just expanding Alamos' comment, $$\mathbb{E}[X^k]=\int_{0}^{1}x^k\,dx=\frac{1}{k+1},$$ and assuming that $X$ and $Y$ are independent, $$\mathbb{E}[XY^k]=\mathbb{E}[X]\cdot\mathbb{E}[Y^k]=\frac{1}{2k+2}.$$