Statistical Theory of a Scaling Circuit

The over-all efficiency of a counter circuit embodying a scaling circuit to reduce the rate of recording is investigated. The probability of detecting counts (G(n, τ)) of a scaling circuit and recorder fed with random impulses is shown to be: 1-I(μτ, n), where n is the scale of the circuit, τ the resolving time of the recorder, μ the statistical average of incoming counts, and where I(μτ, n) is the ratio of the incomplete gamma-function with upper limit μτ of n to the complete gamma-function of n. Efficiency curves (100 G(n, τ)) are plotted showing that the distribution in time of the impulses leaving the scaling circuit is no longer random, but is a distribution in which the extreme short and long intervals between impulses are averaged out. The average rate of recording counts (μₙ) is (μn)G(n, τ). The analysis is extended to include the effect of the finite resolving time of the amplifier (σ), yielding when ${τ}>~n{σ}:{{μ}}ₙ=({{μ}}{n})[{1}{({μ}{σ}+1)}][1{-}I({μ}({τ}{-}n{σ}), n)]$, and when ${τ}<~n{σ}:{{μ}}ₙ=({{μ}}{n})[{1}{({μ}{σ}+1)}]$, the latter being the response of the amplifier alone. Thus, one need only make $n>{{τ}}{{σ}}$ and the scaling circuit and recorder will operate with 100 percent efficiency.