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

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

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

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

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

(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/

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

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