Counting statistics, uncertainty and dimensional models
GRE Subject Test · GRE Physics Topic 21 6:16 English narration · English + 中文 subtitles burned in
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A detector with ten percent efficiency does not register exactly ten photons out of every hundred: the count fluctuates even when the efficiency is stable.
效率为百分之十的探测器并不会在每一百个光子中恰好记录十个：即使效率稳定，计数也会涨落。
This topic uses binomial and Poisson means and fluctuations, propagates small uncertainties with stated correlation assumptions, and solves dimensional exponent constraints to check a model's units.
本主题使用二项分布和泊松分布的均值与涨落，在说明相关性假设的前提下传递小的不确定度，并用量纲指数约束检验模型的单位。
For N independent incident particles each detected with probability p, the count X is binomial, with mean N p and variance N p times 1 minus p.
对于N个独立入射粒子，每个被探测到的概率为p，计数X服从二项分布：均值为Np，方差为Np乘以(1减p)。
The standard deviation is the square root of N p times 1 minus p, not the variance itself.
标准差是Np乘以(1减p)的平方根，而非方差本身。
With large N and small p, a Poisson approximation has mean lambda equal to N p and variance lambda, so the deviation is the square root of lambda.
当N较大且p较小时，泊松近似的均值lambda等于Np，方差为lambda，因此偏差为lambda的平方根。
For N equal to 200 and p equal to 0.05, the exact mean is 10 and the deviation is the square root of 9.5, about 3.08, while Poisson gives the square root of 10, about 3.16.
当N等于200、p等于0.05时，精确均值为10，偏差为9.5的平方根，约等于3.08；而泊松近似给出10的平方根，约等于3.16。
Neither model promises a fixed count.
两种模型都无法保证固定计数。
Correlated detections or dead time can invalidate independence.
相关探测或死时间可能破坏独立性假设。
For a smooth measured function y of x sub one, and so on, linearise changes using its partial derivatives.
对于光滑的测量函数 y(x₁, …)，利用其偏导数对变化进行线性化。
Independent small standard uncertainties combine in quadrature: sigma sub y squared is approximately equal to the sum of the partial derivative of y with respect to x sub i squared, times sigma sub i squared.
独立的微小标准不确定度按平方和合成：σ_y² ≈ Σ(∂y/∂x_i)² σ_i²。
Correlated inputs require covariance cross terms.
相关输入需要协方差交叉项。
For y equals x to the power a, fractional standard uncertainty is approximately equal to the absolute value of a, times sigma sub x, divided by the absolute value of x.
对于 y = x^a，相对标准不确定度约为 |a|σ_x / |x|。
Thus kinetic energy K equals one half m v squared, with negligible mass uncertainty, has fractional uncertainty twice that of speed.
因此，动能 K = ½mv² 在质量不确定度可忽略时，其相对不确定度是速度相对不确定度的两倍。
With independent mass uncertainty, combine sigma sub m over m squared plus two sigma sub v over v squared.
若质量不确定度独立，则合并 (σ_m/m)² + (2σ_v/v)²。
These approximations need small errors and a suitable local linear model.
这些近似需要小误差和合适的局部线性模型。
Random scatter measures precision; a common calibration bias affects accuracy and does not disappear by averaging repeats.
随机离散度衡量精密度。 常见的校准偏差会影响准确度，且无法通过重复测量取平均来消除。
For independent repeated readings, the standard uncertainty of the mean falls as 1/sqrt(n), but shared systematic error does not.
对于独立的重复读数，均值的标准不确定度随一除以根号n而减小，但共享的系统误差不会减小。
The uncertainty of a physical spread and uncertainty of its estimated mean are different quantities.
物理分布的不确定度与其估计均值的不确定度是不同的量。
State whether a quoted percentage is a standard uncertainty, confidence interval or worst-case bound.
请说明所引用的百分比是标准不确定度、置信区间还是最坏情况界限。
Summing absolute contributions is a conservative bound, not the independent-standard-error quadrature rule.
绝对贡献求和是一种保守界限，而非独立标准误差的平方和开方规则。
Dimensional analysis equates powers of mass, length and time, rather than numerical sizes.
量纲分析等式的是质量、长度和时间幂次，而非数值大小。
For a Planck-length form G to the power a, times h-bar to the power b, times c to the power d, dimensions are: G equals L cubed over M T squared; h-bar equals M L squared over T; and c equals L over T.
对于普朗克长度的形式G的a次方乘以h-bar的b次方乘以c的d次方，其量纲为：G等于L立方除以M T平方；h-bar等于M L平方除以T；c等于L除以T。
Requiring length gives minus a plus b equals 0, 3a plus 2b plus d equals 1, and minus 2a minus b minus d equals 0.
要求结果为长度时，得到负a加b等于0，3a加2b加d等于1，以及负2a减b减d等于0。
Hence a equals b equals one half, and d equals negative three halves, so length is the square root of G h-bar divided by c cubed.
因此a等于b等于二分之一，d等于负二分之三，所以长度等于根号下G h-bar除以c立方。
Dimensional analysis cannot determine an arbitrary dimensionless coefficient or prove that the chosen constants are physically sufficient.
量纲分析无法确定任意的无量纲系数，也无法证明所选常数在物理上已足够充分。
Reject a formula with wrong units before inserting numbers.
在代入数值之前，应先排除单位错误的公式。
For K equals one half m v squared with independent small uncertainties, the fractional uncertainty squared is the mass fraction squared plus twice the speed fraction, squared.
对于 K 等于二分之一 m v 的平方，若各不确定度独立且很小，相对不确定度的平方等于质量相对不确定度的平方，加上两倍速度相对不确定度的平方。
With 3 percent in mass and 2 percent in speed, the contributions are 3 and 4 percent, so the result is the square root of 9 plus 16: 5 percent.
质量有百分之 3、速度有百分之 2 的不确定度时，两项贡献分别是百分之 3 和百分之 4，所以结果是 9 加 16 的平方根：百分之 5。
A worst-case sum would give 7.
最坏情况的直接相加会得到百分之 7。
A sample interval contains 900 counts.
一个样本区间包含 900 个计数。
An independent equal-duration background interval contains 100.
一个独立的等时长背景区间包含 100 个计数。
Estimate net counts and their standard uncertainty.
请估算净计数及其标准不确定度。
If both intervals are doubled at stable rates, what happens to relative uncertainty?
如果两个区间的计数率保持稳定，且时间均加倍，相对不确定度会发生什么变化？
This question is worth four marks.
本题共 4 分。
Net count is 800.
净计数为800。
Independent Poisson variances add despite subtraction: sigma equals the square root of 900 plus 100, which equals the square root of 1000, or 31.62 counts, relative uncertainty 0.03953.
独立泊松方差在相减时依然相加：sigma等于根号下900加100，即根号下1000，或31.62个计数，相对不确定度为0.03953。
Doubled intervals give net 1600 and variance 2000, so relative uncertainty is divided by the square root of 2.
时间间隔加倍后净计数为1600，方差为2000，因此相对不确定度除以根号2。
Award net, variance addition, uncertainty and time scaling.
评分要点包括净计数、方差相加、不确定度和时间缩放。
This assumes stable rates and no dead-time correlations.
此计算假设计数率稳定且无死时间关联。
Do not confuse mean count with guaranteed count, variance with standard deviation, or a random standard error with a calibration bias.
请勿将平均计数与保证计数混淆，将方差与标准差混淆，或将随机标准误差与校准偏差混淆。
Dimensional consistency is necessary but not sufficient.
量纲一致性是必要条件，但并非充分条件。
standard uncertainty: Uncertainty expressed as a standard deviation under a stated measurement model.
标准不确定度是指在给定的测量模型下，以标准差表示的不确定度。
Poisson approximation: Rare independent-event count model with variance equal to its mean.
泊松近似是一种稀有独立事件计数模型，其方差等于均值。
Retrieve the meanings without reading the definitions, then change a parameter in the original transfer and test the limiting case.
在不阅读定义的情况下回忆其含义，然后改变原始传递中的一个参数并检验极限情况。