I have been given a Gaussian-distributed random noise n(t) which has an average power of 5 mW. How can I calculate the mean and variance of noise n(t)? Any suggestions? Thanks!
If the values for each sample of $n(t)$ are given, then the mean and variance can be estimated using the equations for sample mean and sample variance, which is trivial so I assume the OP is only given that it is Gaussian-distributed with average power of 5 mW. From that alone, there is no way to know what the mean of the signal is since it hasn't been specified that the process is "zero-mean" or not. If the signal does have a mean value, it will contribute to the total power so this must be specified.
The variance is the portion of the power that is not included in the mean. Since units of power were provided, then the resistance would need to also be given in order to provide units for any non-zero mean (voltage or current) if there is a mean value.