1. Moments of probability distributions
The moments of a probability distribution are a way of characterising its position and shape.
Strong physical analogy with moments in mechanics of rigid bodies
Centre of gravity
Moment of inertia
Higher moments

2. Mean and median Mean value (centre of gravity)
<x>
Mean value (centre of gravity)
Median value (50th percentile)
f(x) x
F(x) 1
1/2
xmed x

3. Variance and standard deviation
Standard deviation  measures width of distribution.
Variance  (moment of inertia)
<x>
f(x) - + x

4. Example: Gaussian distribution G(,2)
Also known as a normal distribution.
Physical example: thermal Doppler broadening
Mean value: <x> = 
Variance: x
Full width at half maximum value (FWHM)
32% probability that a value lies outside  ±
4.5% probability a value lies outside  ±2
0.3% probability a value lies outside  ±3 
f(x)
-
 + x

8. Higher central moments
General form:
e.g. Skewness (m3):
e.g. Kurtosis (m4):
f(x) x
f(x)
Peaky
Boxy x

9. (Pathological) example: Lorentzian (Cauchy) distribution
Physical example: damping wings of spectral lines.
Wings are so wide that no moments converge!
f(x)
x/
F(x)
x/

10. Poisson distribution P()
Bin number
Counts per bin
 = 5
A discrete distribution
Describes counting statistics:
Raindrops in bucket per time interval
Cars on road per time interval
Photons per pixel during exposure
 = mean count rate P 1 2 4 8  x

11. Exponential distribution
Distribution of time intervals between events
Raindrops, cars, photons etc
A continuous distribution