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This article theoretically considers how the fundamental mechanism of a mean-squaring operation existing in sound level or vibration level meters affects the probability distribution form of the output level fluctuation in direct connection with the probabilistic evaluation problem, by applying the well-known residue theorem to its moment generating function. More specifically, the explicit expression for the output level distribution is theoretically proposed in a general expansion form, after introducing the so-called Parseval's relation at the starting point of theoretical analysis, in the case when a Gaussian random wave with arbitrary frequency characteristic is measured through usual sound or vibration level meters. The effect of the time constant and the frequency characteristic of the indicating system and the frequency spectrum of input random noise on the output probability distribution is concretely reflected in each parameter of the output probability distribution expression. The validity of the above theoretical result is confirmed experimentally by applying it to the actual random noise data observed by a usual sound level meter.

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