Conditional expectation of iid nonnegative random variables I am studying Ross's book, stochastic processes. There is the following lemma:

Let $Y_1, Y_2, ... , Y_n$ be iid nonnegative random variable. Then,
$E[Y_1+ \cdots +Y_k | Y_1+\cdots+Y_n=y] =  \frac{k}{n} \cdot y, \quad k=1,\cdots,n$

But, I really can't understand why this lemma can be established. Could you please help me?

Informally, as all the $Y_i$ are identically distributed, the fact that they sum to $y$ implies that their (conditional) expectation must be $\frac yn$.  To see this formally, note that we certainly have equality of all the $E[Y_i\;|\;Y_1+...+Y_n=y]$, call the common value $E$.  Then
$$nE=E\left[Y_1\;|\;Y_1+Y_2+...+Y_n=y\right]+...+E\left[Y_n\;|\;Y_1+Y_2+...+Y_n=y\right]=E\left[Y_1+...+Y_n\;|\;Y_1+...+Y_n=y\right]=y$$
Thus $E=\frac yn$.
But then $$E\left[Y_1+...+Y_k\;|\;Y_1+Y_2+...+Y_n=y\right]=kE=\frac {ky}{n}$$
As desired.
