# Variance in a known population

I am working on a problem that is similar to the standard radioactive decay rate experiment, but with a twist. In the normal experiment, one takes several different measurements of the decay rate, then computes the decay rate (or time) and the associated error. My background is in high energy physics, so while I am used to counting experiments, I am having issues with the following:

I have a twist on the same problem. I have $N$ objects which can be in state $a$ or $b$, and the number in each state is $N_a$ and $N_b$, with $N = N_a + N_b$. The system is prepared so that we know that every object is in state $a$ to begin with.

Then, I am interested in the number of objects in state a with respect to time (I will be fitting it later). Naively, I would expect the error to be just the $\sqrt{N_a}$, except that I perfectly know the system starts off in state $a$, which my intuition tells me that I would have an error of 0 on that point.

Since there could be an error in $N_a$ as well as $N_b$ (except for the dependence), I was also looking at calculating the error (assuming the errors in $N_a$ and $N_b$ are independent) using a ratio:

$$N = N_a + N_b;N(0)=N_a$$

$$f(N_a,N_b) = \frac{N_a}{N_a+N_b}$$

$$\delta f = \sqrt{\frac{f(1-f)}{N}}$$

Which makes more sense to me, since the error goes to zero at the two bounds when we know that we have reached either all objects in state $a$ or in state $b$.

Could somebody direct me to a paper, or a description of what I am looking for? I am sure that this particular problem has been solved and resolved, but I do not know enough of the proper terms to get it. As well, what would people recommend for a nice, thorough treatment of calculating errors at the graduate level?

EDIT:Additional Information So after thinking about it, this isn't quite what I am looking for. Let me describe what I am actually measuring, and then maybe I can get a start in the right direction.

I run N simulations, each of which is prepared in state a. Then, I record the time when it switches from a->b. This is irreversible, you can't go from b->a. In the end, I have my N simulations, each with a time t. Then I plot the fraction remaining in state a at time t, call it $f(a,t)$, binning time.

Now, this fraction cannot be the PDF, since it is not normalized to 1. What I want out of the distribution is the decay rate (or time) of a given object. I also want the statement where I can look at an object and ask: What is the probability that it will be in state a at time t? Isn't that just my original distribution? I also want the error on each time bin, as well as the error on the variables in the fit (I am using MATLAB and nonlinear regression for now, so that is easy).

I realize that I might have some sort of fundamental misunderstanding of some of this, so please, let me know what I can do to improve my questions/answers.

• Much of what you discuss here can be conceptualized through the binomial distribution, so that may be something to start exploring. However, I think what you ultimately want is to understand / model how the parameter, $p$, underlying the binomial distribution is changing over time. – gung May 30 '13 at 3:53
• I'd second @gung's mention of the binomial (/bernoulli) model; there are a variety of tools for modelling such responses over time. If you're trying to model changes of state over time, you might want to look at things like Markov chains as well (those aren't competing suggestions, but complementary ones, dealing with different aspects of the problem). – Glen_b May 30 '13 at 4:13
• Ok, thanks! The binomial distribution link helped, and confirms the variance that I am using. Thanks both! – NuclearAlchemist May 30 '13 at 17:42
• Updated the main question with additional information. – NuclearAlchemist Jun 7 '13 at 18:52
• It may mislead you--and others--to refer to $a$ as the "state." If this is indeed a full description of the physical state, then the decay rate cannot depend on time, leading to the usual exponential curve. Otherwise, the decay rate does depend on the time a particle has been classified as "$a$", whence the true state at the very least requires two values: the label $a$ or $b$ together with the amount of time it has had that label (if it is labeled $a$). – whuber Jun 7 '13 at 22:57