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:
- If the number of clicks per second is less than 4, there should be only l chance in 500 that the alarm will sound.
- 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?
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.