Let a source of thermal noise be connected to the input of a band pass filter. Consider the energy which would be dissipated during the interval t1 to t1 + T if the output current were to flow through a resistance of one ohm. When T is held fixed and t1 regarded as a random variable, the resulting energies have a distribution whose average and standard deviation depend upon T. Here this dependence is studied. The standard deviation of the difference of the energies of two contiguous intervals, each of length T, is also obtained.