Statistical Theory of a Scaling Circuit

Abstract

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 ($G(n, \tau )$) of a scaling circuit and recorder fed with random impulses is shown to be: $1-I(\mu \tau , n)$, where $n$ is the scale of the circuit, $\tau$ the resolving time of the recorder, $\mu$ the statistical average of incoming counts, and where $I(\mu \tau , n)$ is the ratio of the incomplete gamma-function with upper limit $\mu \tau$ of $n$ to the complete gamma-function of $n$. Efficiency curves (100 $G(n, \tau )$) 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 (${\mu }_n$) is $\left(\frac{\mu }{n}\right)G(n, \tau )$. The analysis is extended to include the effect of the finite resolving time of the amplifier ($\sigma$), yielding when $\tau >~n\sigma :{\mu }_n=\left(\frac{\mu }{n}\right)\left[\frac{1}{(\mu \sigma +1)}\right][1-I(\mu (\tau -n\sigma ), n\left)\right]$, and when $\tau <~n\sigma :{\mu }_n=\left(\frac{\mu }{n}\right)\left[\frac{1}{(\mu \sigma +1)}\right]$, the latter being the response of the amplifier alone. Thus, one need only make $n>\mathrm{}\frac{\tau }{\sigma }$ and the scaling circuit and recorder will operate with 100 percent efficiency.