# Uncertainties in frequencies

Suppose we have some radioactive process and suppose we want to measure how many particles are emitted by the radioactive material in a given time (say $t=10 \text{s}$). Suppose we did $N=150$ measurements and got the following results:

where "Value" ($n$) represents the amount of particles hitting the detector in $t=10 \text{s}$.

Since we have the frequency ($f(n)$) for each value, we can calculate the "probability": $P=f(n)/N$.

Now since we suspect that this distribution is Poisson, we would like to search for the optimal $\lambda$ such that:

$$P=\frac{\lambda^n}{n!}e^{-\lambda}$$

Now, the Chi-squared test (minimizing chi-squared) requires us to know the uncertainties in our variables. In many texts there's a reasonable assumption that the uncertainty in $n$ should be equal to the standard deviation of $\text{unif}(0,1)$, that is $\Delta n=1/\sqrt{12}$. However I have no idea what should be the uncertainty of the value $P$. Obviously, we must have $\Delta P=(\Delta f(n)) / N$. But what should be the uncertainty of the frequency? (I don't think that it should depend on the precision of the detector).

• There seems to be some confusion expressed in this question: the chi-squared test does not minimize a chi-square statistic. It merely computes it and compares it to quantiles of a suitable chi-square distribution. The "reasonable assumption" about the uncertainties makes no sense in this context. The characterization of an optimal $\lambda$ makes little sense, either, because it corresponds to an infinite set of simultaneous equalities for $n=0,1,2,\ldots$. Could you clarify what you mean by "uncertainty of the frequency"? What frequency, specifically?
– whuber
Nov 19 '17 at 17:28
• @whuber In our physics lab we learned the following: suppose we got N measurements: $(x_1,y_1),(x_2,y_2),...,(x_N,y_N)$ and suppose each measurement has some uncertainty $\sigma_i$. Now suppose we have some model (function) $f(\vec{a};x_i)$ that we believe describes the data the best. Then we would like to find parameters $\vec{a}$ that minimize $\chi^2=\sum_{i=1}^{N}\left(\frac{y_i-f(\vec{a};x_i)}{\sigma_i}\right)^2$. I believe this answer describes a similar procedure. Nov 19 '17 at 18:01
• @whuber - back to the question "what is uncertainty of the frequency"? In the described experiment we would like to verify that the set of points $(n,f(n)/N)$ can be described by a Poisson distribution. But in order to determine $\sigma_i$ we must know the uncertainties in the measured value $n$ and in the value of $f(n)/N$ (which is the relative frequency). "Frequency" here means the number of times we got a specific value of $n$ in this experiment. For example, if we got the value $n=44$ twelve times then $f(n=44)=12$ and the relative frequency is $f(n)/N=12/150$ (here $N=150$) Nov 19 '17 at 18:07
• Your description is rather complex and unnatural. Aren't you just trying to estimate the parameter of a Poisson distribution that might describe these data? Least squares, which is what you seem to mean by "chi-squared," isn't even applicable here.
– whuber
Nov 19 '17 at 22:24
• @whuber - as I understand it - yes - we want to estimate the parameter of a Poisson distribution. We learned that such process of estimation must maximize the likelihood $L$. But since the likelihood satisfies the relation $\ln L = \text{const} - \frac{1}{2} \chi^2$ it's equivalent to minimizing $\chi^2$. Sorry if I sound confused (that's how poorly physicists are taught statistics). Nov 20 '17 at 11:16