Compute $E[S_n\mid T_1+\cdots+T_n\leq w<T_1+\cdots+T_{n+1}]$, where $S_n=\sup\{T_1,r\}+\cdots+\sup\{T_n,r\}$

Question

Let $(T_i)$ denote a sequence of iid exponential random variables with parameter $\lambda$, $r$ some constant, and, for every $i$, $X_i=\sup\{T_i,r\}$. How to compute $$E[X_1+X_2+\cdots+X_n\mid T_1+\cdots+T_n\leq w<T_1+\cdots +T_{n+1}]$$ for some given $w$?

I was able to find an expression for this if $X_i$ is simply an exponential random variable, but am having trouble if $X_i$ is as defined above.

Answer 1

$X_n$'s are independent. Hence conditioning on $\{X_{n+1} >w\}$ has no effect. So the value is $E(\sum_{i=1}^{n} X_i |X_n \leq w)=\sum_{i=1}^{n-1} EX_i+E(X_n|X_n \leq w)$. I suppose you can take it from here.