The probability density function of the envelope of narrow band Gaussian noise is

The amplitude spectrum of a Gaussian pulse is

A probability density function is given by $$p(x) = \,K\,\,\exp \,\,( - \,{x^2}/2),\,\, - \,\infty \, < \,x < \,\infty $$.

The value of K should be

For a narrow band noise with Gaussian quadrature and inphase components, the probability density function of its enveolope will be