Originally Posted by

**matheagle** Back to $\displaystyle E(I(X_1<c)|S_n=s)$ where s>c.

We need the density $\displaystyle f(x_1|s_n) ={f(x_1) f(s_{n-1}) \over f(s_n) }$, where $\displaystyle s_{n-1}=x_2+\cdots +x_n$.

The density of $\displaystyle X_1$ is $\displaystyle f(x_1)=e^{-x_1/q}/q$.

While $\displaystyle f(s_{n-1})= {s_{n-1}^{n-2} e^{-s_{n-1}/q}\over \Gamma(n-1)q^{n-1}}$ and $\displaystyle f(s_n) ={s_n^{n-1} e^{-s_n/q}\over \Gamma(n)q^n}$

Noting that $\displaystyle x_1+s_{n-1}=s_n$ we have

$\displaystyle f(x_1|s_n) ={(n-1) (x_2+\cdots +x_n)^{n-2} \over (x_1+\cdots +x_n)^{n-1}}$

Thus $\displaystyle P(X_1<c|S_n=s)=\int_0^cf(x_1|S_n=s)dx_1=(n-1)\int_0^c {(s-x_1)^{n-2}dx_1\over s^{n-1}}$

which gives me $\displaystyle 1-\biggl(1-{c\over s}\biggr)^{n-1}$, which doesn't make sense when n=1.