To begin you must write the output of filter as a convolution (read a book on stochastic process or stochastic signal processing).
Your new random variable which is the output of filter is a linear combination of filter input which has Gaussian distribution, also we know linear combination of gaussian RV is another gaussian RV.
Now taking expectation from both side of convolution, we see expectation of your new random variable (its mean) is DC response of your filter (which is usually 1) times the mean of your old random variable.
To obtain the variance you have to subtract the mean from from the signal then convolve it with its reversed time version then taking expectation at time zero. At the end you see new RV variance is the variance of input times square of DC response of filter.
I have to say it was better to use random procedd instead of random variable.