Given the power spectral density, how do I calculate the bandwidth and total power of \$X(t)\$?
For noise, its power can be computed as the product of the noise power spectral density and the transmission bandwidth. It is important to note that as you make more room for your signal, meaning, more transmission bandwidth for your signal, the noise power also increases owing to the linear relation between the noise power and the signal transmission bandwidth. This is followed in noise modelling systems including AWGN model.