A large system with property [itex]p[/itex] is observed many times. The average of this property after [itex]N[/itex] observations is(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\langle p \rangle_N = \frac{1}{N}\sum^N_{i=1}p_i[/tex]

where [itex]p_i[/itex] is the [itex]i[/itex]th observation.

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

(b) More generally, supposing that all the measurements are uncorrelated, show the variance per measurement, [itex]\langle [p-\langle p \rangle_{\infty}]^{2} \rangle_{\infty}/N[/itex], is independent of [itex]N[/itex] and determine the uncertainty in [itex]p[/itex], 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 [itex]\langle [\langle p \rangle_{N} - \langle p \rangle_{\infty}]^2 \rangle^\frac{1}{2}[/itex], but I'm really not sure where to start with this. I believe that [itex]\langle p \rangle_\infty = x_a a + x_b b + (1-x_a-x_b) c[/itex] but I'm not sure how this helps (especially since the answer should be in terms of [itex]\langle p \rangle_\infty[/itex]).

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!

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# Observing a system N times

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