Статистика подсчета, неопределенность и размерные модели
Введено| English | Значение |
|---|---|
| Poisson approximation/ˈpɔɪsn əˌprɒksɪˈmeɪʃn/ | приближение Пуассона |
| standard uncertainty/ˈstændəd ʌnˈsɜːtənti/ | стандартная неопределённость |
A decision before an answer
- A detector with ten percent efficiency does not register exactly ten photons out of every hundred. The count fluctuates even with stable efficiency.
- Your goal: Use binomial and Poisson count means and fluctuations.
Read the relationship
- For N independent incident particles each detected with probability p, count X is binomial: mean Np and variance Np(1−p). The standard deviation is sqrt[Np(1−p)], not the variance itself. With large N and small p, a Poisson approximation has mean λ=Np and variance λ, so deviation sqrt(λ). For N=200 and p=0.05, the exact mean is 10 and deviation sqrt(9.5)≈3.08, while Poisson gives sqrt(10)≈3.16. Neither model promises a fixed count. Correlated detections or dead time can invalidate independence.
- Propagate small uncertainties with stated correlation assumptions.
A binomial detector has N=100 and p=0.20. Its mean and standard deviation are:
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Mean 20; variance 100·0.2·0.8=16, so standard deviation 4.
Use the defining rule
- For a smooth measured function y(x1,…), linearise changes using its partial derivatives. Independent small standard uncertainties combine in quadrature: σ_y²≈Σ(∂y/∂xi)²σ_i². Correlated inputs require covariance cross terms. For y=x^a, fractional standard uncertainty is approximately |a|σ_x/|x|. Thus kinetic energy K=½mv² with negligible mass uncertainty has fractional uncertainty twice that of speed. With independent mass uncertainty, combine (σ_m/m)²+(2σ_v/v)². These approximations need small errors and a suitable local linear model.
- Solve dimensional exponent constraints and check a model’s units.
With negligible mass uncertainty and speed uncertainty 3%, small kinetic-energy fractional uncertainty is:
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K depends on v², so the fractional uncertainty doubles.
Check the conditions
- Random scatter measures precision; a common calibration bias affects accuracy and does not disappear by averaging repeats. For independent repeated readings, the standard uncertainty of the mean falls as 1/sqrt(n), but shared systematic error does not. The uncertainty of a physical spread and uncertainty of its estimated mean are different quantities. State whether a quoted percentage is a standard uncertainty, confidence interval or worst-case bound. Summing absolute contributions is a conservative bound, not the independent-standard-error quadrature rule.
- Solve dimensional exponent constraints and check a model’s units.
For N=200 and p=0.05, expected detector count is 10 with standard deviation sqrt(9.5). If speed has 4% small standard uncertainty and mass uncertainty is negligible, kinetic-energy uncertainty is approximately 8%. The dimensional exponents (1/2,1/2,−3/2) give zero mass power, one length power and zero time power for sqrt(Gℏ/c³).
A Poisson count has mean 25. Its standard deviation is ____.
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Poisson variance equals mean, and standard deviation is its square root.
Apply the task format
- Dimensional analysis equates powers of mass, length and time, rather than numerical sizes. For a Planck-length form G^a ℏ^b c^d, dimensions are [G]=L³/(MT²), [ℏ]=ML²/T and [c]=L/T. Requiring length gives −a+b=0, 3a+2b+d=1 and −2a−b−d=0. Hence a=b=1/2 and d=−3/2, so length is sqrt(Gℏ/c³). Dimensional analysis cannot determine an arbitrary dimensionless coefficient or prove that the chosen constants are physically sufficient. Reject a formula with wrong units before inserting numbers.
- Solve dimensional exponent constraints and check a model’s units.
Do not confuse mean count with guaranteed count, variance with standard deviation, or a random standard error with a calibration bias. Dimensional consistency is necessary but not sufficient.
Какой ответ подходит к данному случаю?
Использовать средние значения и флуктуации биномиального и пуассоновского счета
A dimensionally consistent formula necessarily has the correct numerical coefficient and physical assumptions.
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Units do not determine dimensionless factors or validate the physical model.
Keep the distinctions
- standard uncertainty 标准不确定度 — Uncertainty expressed as a standard deviation under a stated measurement model.
- Poisson approximation 泊松近似 — Rare independent-event count model with variance equal to its mean.
- Use binomial and Poisson count means and fluctuations.
- Propagate small uncertainties with stated correlation assumptions.
- Solve dimensional exponent constraints and check a model’s units.
Match each term with its precise meaning in this lesson.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.