Probability and random variables
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| variance/ˈveərɪəns/ | variance |
Probability: variances add, σ/√n averages away
- Variance 方差 $E[X^{2}] - \mu ^{2}$; for independent sources variances ADD (σ does not). Gaussian bands: ±1σ 68%, ±2σ 95%.
- Averaging n independent noise samples divides σ by √n — 16 averages turn 2 mV into 0.5 mV.
{exercise:uni-ee2x-prob}
Two independent noise sources, σ = 2 mV and 3 mV, add. The combined σ is —
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Independent sources add in quadrature: √(4 + 9) = √13. Only perfectly correlated noise would add σ's linearly.
Two independent noise sources, 2 mV and 3 mV rms. The combined noise is —
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Independent variances add: sqrt(2^2 + 3^2) = sqrt(13) = 3.61 mV - smaller than 2+3 = 5 because deviations do not peak together. Halving the dominant 3 mV source to 1.5 mV would give sqrt(1.5^2+2^2) = 2.5 mV, a 3.2 dB win.
Averaging 16 independent noise readings improves SNR by about —
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sigma falls by sqrt(16) = 4, so the signal-to-noise ratio rises 4x = 12.0 dB - but only for independent noise: mains hum is correlated and refuses to average away.