Deriving statistics of band limited Random Noise
at first read a book on stochastic process or stochastic signal processing.
it's better that we say we have a random process which it's first order distribution (PDF of the process at each instant of time) is independent of time and is a Gaussian with known mean and variance. Now we want to obtain the first order properties of filtered process.

To begin we write the output of filter as a convolution. The output process at each instant of time is a linear combination (weighted average) of input process at different instant of time which all has the same Gaussian distribution, also we know linear combination of variables with Gaussian PDF leads to another variable with Gaussian distribution. So the first order distribution of output must be a Gaussian.

Now taking expectation from both side of convolution, we see expectation of filtered process at each instant of time becomes convolution of filter response and the mean of input process which is a constant over time, so it becomes the multiplication of input mean and the DC response of filter. 
To obtain the variance of process around its mean, we have to find the expectation of squared difference of the process and its mean or expectation of squared value of output minus the square of it's mean. To relate the squared value of output to the input we could convolve the filterd signal with its reversed time version at zero lag. At the end we see new variance is the variance of input times square of DC response of filter.