Let $Y_i$ be a sequence i.i.d random variables. I define $X = \frac{1}{\sum_{i=1}^{k} Y_i} .$ Now, I'd like to calculate $E[Y_1X]$. Intuitively, this is just $1/k$ which seems to suggest $Y_1$ and $X$ are independent however, value of $X$ certainly depends on $Y_1$.
So, are X and $Y_i$ are independent or am I confusing things here? In general, what's the correct way to compute expectation here given that I know the distribution of $Y$ ?
Firstly, as Arthur says, the relation $\mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y]$ doesn't imply independence: Take any symmetric $X\in L^3$, then $\mathbb{E}[X\cdot X^2]=0=\mathbb{E}[X]\mathbb{E}[X^2]$.
Secondly, your intuition is correct: In order to compute $\mathbb{E}[Y_1 X]$ in your situation, the central observation is that the $Y_i X$, $i\in\{1,2,\ldots,k\}$, are identically distributed. Now you have $\sum_{i=1}^n Y_i X=1$, so that $$1=\mathbb{E}\left[\sum_{i=1}^k Y_i X\right]=\sum_{i=1}^k\mathbb{E}\left[ Y_i X\right]=k\mathbb{E}[Y_1 X].$$