Let $N_t$ be a Poisson process with rate $\lambda$.
$T_k$ the inter arrival times of $N_t$.
$\{X_k\}$ a collection of i.i.d. random variables with mean $\mu$.
$X_k$ is independent of $N_t$.
Calculate the expectation of $ S_t= \sum_{k=1}^{N_t} X_k e^{t-T_k}. $
Given $N_t$, the inter arrival times are uniformly distributed on $[0,t]$.
Hence, $T_k \sim \text{Beta}(k,n-k+1)$ and $ E\left( \left. e^{-T_k}\right| N_t=n \right)=\frac{1}{B(k,n-k+1)}\int_0^1 e^{-x}x^{k-1} (1-x)^{n-k} dx. $ I don't see how to compute this integral.