1. Statistical properties of Random time series (“noise”)

2. Normal (Gaussian) distribution Probability density: A realization (ensemble element) as a 50 point “time series” Another realization with 500 points (or 10 elements of an ensemble)

3. From time series to Gaussian parameters N=50: =5.57 (11%); ) 2 >=3.10 N=500: =6.23 (4%); ) 2 >=3.03 N=10 4 : =6.05 (0.8%); ) 2 >=3.06

4. Divide and conquer Treat N=10 4 points as 20 sets of 500 points Calculate: – mean of means: E{  =5.97 – std of means:   = =0.13 Compare with – N=500: =6.23; =3.03 – N=10 4 : =6.05; =3.06 – 1/√500=0.04; 2    E{ 

5. Generic definitions (for any kind of ergodic, stationary noise) Auto-correlation function For normal distributions:

6. Autocorrelation function of a normal distribution (boring)

8. Frequency domain Fourier transform (“FFT” nowadays): Not true for random noise! Define (two sided) power spectral density using autocorrelation function: One sided psd: only for f >0, twice as above. IF

10. Discrete and finite time series 

11. Take a time series of total time T, with sampling  t Divide it in N segments of length T/N Calculate FT of each segment, for  f=N/T Calculate S(f) the average of the ensemble of FTs We can have few long segments (more uncertainty, more frequency resolution), or many short segments (less uncertainty, coarser frequency resolution)