Counting statistics, uncertainty and dimensional models
| English | Español |
|---|---|
| Poisson approximation | Poisson approximation |
| standard uncertainty | standard uncertainty |
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:
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:
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 ____.
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.
Which answer fits this case?
Use binomial and Poisson count means and fluctuations
A dimensionally consistent formula necessarily has the correct numerical coefficient and physical assumptions.
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.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.