When I read Ross's book "Statistic Process" , I find the lemma, but I cannot prove it. The lemma states that Let $Y_1, \cdots, Y_n$ be iid nonnegative random variables then $E[Y_1+\cdots + Y_k| Y_1+\cdots+Y_n = y] =\frac{ky}{n}$, $k=1, .., n$.
I tried to calculate the expectation directly, Let $S_k = Y_1+\cdots + Y_k, k=1, 2, ..., n$, then $E[S_k| S_n = y] = \int_{0}^{y} t P\{S_k= t | S_n = y\}dt = \int_{0}^{y}P\{S_k \in [t, t+\delta], S_{k-1}, .., S_{1} \in [0, t)\}t dt= \int_{0}^{y}n\frac{1}{y} P\{S_{k-1}, .., S_{1} \in [0, t)\} dt= \int_{0}^{y}\frac{n}{y}C_{n-1}^{k-1} (\frac{t}{y})^{k-1}(1-\frac{t}{y})^{n-k}dt$.
I can not calculdate this integration. So I want to look for some other explaination about the result or anyone who can help me calculate the expectation.
We can use the following: for each permutation $\sigma$ from $\{1,\dots,n\}$ to itself, $$\tag{*} \mathbb E\left[Y_1+\dots+Y_k\mid Y_1+\dots+Y_n\right]=\mathbb E\left[Y_{\sigma(1)}+\dots+Y_{\sigma(k)}\mid Y_1+\dots+Y_n\right], $$ which is due to the fact that the vectors $\left(Y_1+\dots+Y_k,Y_1+\dots+Y_n\right)$ and $\left(Y_{\sigma(1)}+\dots+Y_{\sigma(k)}, Y_1+\dots+Y_n\right)$ have the same distribution.
Then sum (*) over all the permutations and rearrange $\sum_{\sigma\in\mathcal S_n}\left(Y_{\sigma(1)}+\dots+Y_{\sigma(k)}\right)$.