Max to Mean Ratio of Normal data

One oversimplified method to look for a valid signal in a frequency band is to first compute the FFT of the signal in the desired frequency band and then ascertain if any point in FFT is crossing a detection threshold. This can be achieved by computing the ratio of the maximum of the FFT output to that of the sample mean value. There are host of other applications that use detection procedure which is very similar to this max-mean ratio test. My task is to find a closed form