I encountered a simple question about computing the following expectation: $$E \left[C^{N_t}\right]$$ where $C>0$ is a fixed constant and $N_t$ is standard Poisson process
My thinking is follows: $$\begin{array}{l} E{C^{{N_t}}} = \sum\limits_{k = 0}^\infty {{C^k}P\left( {{N_t} = k} \right)} = \sum\limits_{k = 0}^\infty {{C^k}{e^{ - \lambda t}}\frac{{{{\left( {\lambda t} \right)}^k}}}{{k!}}} \\ \begin{array}{*{20}{c}} {}&{}&{} \end{array} = {e^{ - \lambda t}}\sum\limits_{k = 0}^\infty {\frac{{{{\left( {C\lambda t} \right)}^k}}}{{k!}}} = {e^{ - \lambda t}}{e^{C\lambda t}} = {e^{\left( {C - 1} \right)\lambda t}} \end{array}$$ The thing bothers me is that $N_t$ is a process but what I'm doing here is simply treat it as a random variable. Is this thinking correct?
Thanks
In general random processes are functions of a parameter where each instance is a random variable. Your thinking is correct.