There are two kind of average power I encountered in random signal class and textbook:
definition 1: average power =$$E[|x(t)|^2]=R_{xx}(0)=\int^\infty_{-\infty} S_{xx}(f)\,df$$ definition 2: average power =$$\lim_{T\to\infty}\frac{1}{2T}\int^{T}_{-T}|x(t)|^2\,dt $$ It seems that the two definition is different unless the signal is ergodic in $x(t)^2$. By the way, our teacher keep using the term "assuming average power of white noise $n(t)$ is $\sigma_N^2$" in test. But from the above definition, the average power is infinite for continuous white noise. I think in discrete white noise the average power can be well-defined but in continuous one I am wondering our teacher may take the definition wrong.