Observing a system N times

[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!

 

[mmwave]

Hello! I can help you with this problem. Let's start with part (a):

First, we can rewrite the expected value \langle p \rangle_N as:
\langle p \rangle_N = \frac{x_a a + x_b b + (1-x_a-x_b) c}{N}

Since we are assuming N to be large and that each observation is uncorrelated with the next, we can use the Central Limit Theorem to approximate the distribution of \langle p \rangle_N as a normal distribution with mean \langle p \rangle_\infty and standard deviation \sigma = \sqrt{\frac{\langle [p-\langle p \rangle_{\infty}]^{2} \rangle_{\infty}}{N}}.

Therefore, the expected size of possible deviations from \langle p \rangle_\infty can be expressed as \sigma = \frac{1}{\sqrt{N}}\sqrt{\langle [p-\langle p \rangle_{\infty}]^{2} \rangle_{\infty}}.

Now, we can calculate the variance \langle [p-\langle p \rangle_{\infty}]^{2} \rangle_{\infty} as follows:

\langle [p-\langle p \rangle_{\infty}]^{2} \rangle_{\infty} = \sum_{i=1}^{N}(p_i - \langle p \rangle_\infty)^2 \cdot P(p_i)

Since there are only three possible values of p, we can substitute them into the above equation and solve for the variance. This will give us the expected size of possible deviations from \langle p \rangle_\infty in terms of \langle p \rangle_\infty and N.

For part (b), we can use the same approach as in part (a) to calculate the variance per measurement. The difference is that now we have to divide the variance by N, since we are looking for the variance per measurement. The uncertainty in p can then be expressed as the square root of this variance per measurement.

I hope this helps! Let me know if you have any further questions.