# Question from Statistics for Experimenters (George Box)

The level of radiation in the control room of a nuclear reactor is to be automatically monitored by a Geiger counter. The monitoring device works as follows:

Every tenth minute the number (frequency) of "clicks" occurring in t seconds is counted automatically. A scheme is required such that if the frequency exceeds a number c an alarm will sound. The scheme should have the following properties:

1. If the number of clicks per second is less than 4, there should be only l chance in 500 that the alarm will sound.
2. If the number reaches 16, there should be only about 1 chance in 500 that the alarm will not sound.

What values should be chosen for t and c? Hint: Utilize the Poisson distribution and normal approximation.

My thoughts so far:

The number of clicks we measure, $${N}$$, in $${t}$$ seconds can be assumed to be approximately Poisson with true rate $${\eta t}$$. With time $${t}$$ being a constant, the scheme relates to $${P(\frac{N}{t}>c)=P(N >ct)}$$. Hence we need to choose constants $${c}$$ and $${t}$$ such that the alarm sounds with probability $${\frac{1}{500}}$$ when the true rate per second $${\eta}$$ is less than 4, and the alarm should sound with probability $${\frac{499}{500}}$$ when the true rate per second is greater than 16. Thus we need $${P(N > ct|\eta<4)=1/500}$$ and $${P(N > ct |\eta \ge 16)=499/500}$$. We can approximate $${N}$$ as $${Normal(\eta t, \sqrt{\eta t})}$$. In the first case we have $${P(Z > \frac{ct-\eta t}{\sqrt{\eta t}})=0.002}$$ and $${\eta < 4}$$. Because we're only given a bound for $${\eta}$$ rather than the true value for $${\eta}$$, I'm at a loss of what to do next. Am I going about this the right way?

• Incidentally, the way the numbers are given, and the nice way they work out, make me think that condition $1$ should really read "if the number of clicks per second equals $4$..." (to match the wording of "reaches" in condition $2$). – Brian Tung Mar 12 at 17:10

Basic approach. The symmetry between the probabilities of triggering and not triggering the alarm suggest a ratio between $$c$$ and $$t$$ of $$m$$ to $$1$$ (it is some kind of mean of $$4$$ and $$16$$). That is to say, we set $$c = mt$$.
How does this work? Suppose the actual rate is $$4$$ clicks per second. Then over a period of $$t$$ seconds, the distribution of the number of clicks is Poisson with mean $$4t$$. For large-ish means $$\lambda$$, the Poisson distribution is roughly normal with variance $$\lambda$$ (and therefore standard deviation $$\sqrt\lambda$$); in this case, that means a mean of $$4t$$ and a standard deviation of $$2\sqrt{t}$$. The threshold of $$c = mt$$ would then be $$\frac{(m-4)t}{2\sqrt{t}} = \frac{m-4}{2}\sqrt{t}$$ standard deviations above the mean.
Similarly, for an actual rate of $$16$$ clicks per second, a threshold of $$c = mt$$ would be $$\frac{(16-m)t}{4\sqrt{t}} = \frac{16-m}{4}\sqrt{t}$$ standard deviations below the mean. By symmetry, those two $$z$$-scores should be of equal magnitude. Find the value of $$m$$ that satisfies this.
Lastly, find the value of $$t$$ for which the common $$z$$-score gives the appropriate probability $$1/500$$ for the mass in the tail.
• Incidentally, I am not sure that for the numbers you actually get, the normal approximation is valid enough to support a tail mass as small as $1/500$. (I don't know it isn't; I'm just not sure.) But they told you to use the normal approximation, so... – Brian Tung Mar 12 at 17:43