# Observing a system N times

1. Sep 6, 2010

### mooshasta

A large system with property $p$ is observed many times. The average of this property after $N$ observations is
$$\langle p \rangle_N = \frac{1}{N}\sum^N_{i=1}p_i$$
where $p_i$ is the $i$th observation.

(a) Suppose only three values of $p$ are possible, $a$, $b$, and $c$, and these values have the probabilities $x_a$, $x_b$, and $x_c = 1 - x_a - x_b$, respectively. Assume $N$ is large and that each observation is uncorrelated with the next. Determine the expected size of possible deviations from $\langle p \rangle_\infty$. Express your result in terms of $\langle p \rangle_\infty$ and $N$.

(b) More generally, supposing that all the measurements are uncorrelated, show the variance per measurement, $\langle [p-\langle p \rangle_{\infty}]^{2} \rangle_{\infty}/N$, is independent of $N$ and determine the uncertainty in $p$, i.e., determine the expected size of possible deviations from the mean value.

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For part (a), I think I need to determine the value of $\langle [\langle p \rangle_{N} - \langle p \rangle_{\infty}]^2 \rangle^\frac{1}{2}$, but I'm really not sure where to start with this. I believe that $\langle p \rangle_\infty = x_a a + x_b b + (1-x_a-x_b) c$ but I'm not sure how this helps (especially since the answer should be in terms of $\langle p \rangle_\infty$).

For (b) I'm pretty lost as well. Maybe if I can get a handle on part (a), it will be easier.

Thanks in advance for any pointers or hints on where to get started!