Filtering random Signal
The output signal will still be normally distributed, but its power spectrum, i.e. its frequency content, will obviously be different from the input signal. If $S_X(\omega)$ is the power spectrum of the input signal, which is approximately flat, then the power spectrum of the output signal is

$$S_Y(\omega)=|H(\omega)|^2S_X(\omega)$$

where $H(\omega)$ is the frequency response of the filter.