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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) f(x) <x> x f (x) x dx • Median value (50th percentile) 1 F(x med ) 2 P(x xmed ) P(x x med ) x F(x) 1 1/2 0 xmed x Variance and standard deviation • Standard deviation measures width of distribution. • Variance (moment of inertia) 2 = Var(x) [x x ]2 f(x) <x> f (x) [x x ]2 dx - + x Example: Gaussian distribution G(,2) • Also known as a normal distribution. • Physical example: thermal Doppler broadening 1 f (x) e 2 1 x 2 2 f(x) • Mean value: <x> = • Variance: x - • Full width at half maximum value (FWHM) FWHM 8 ln 2 2.3 • 32% probability that a value lies outside ± • 4.5% probability a value lies outside ±2 • 0.3% probability a value lies outside ±3 + x Higher central moments • General form: mn x x • e.g. Skewness (m3): • e.g. Kurtosis (m4): n f(x) x f(x) Peaky Boxy x (Pathological) example: Lorentzian (Cauchy) distribution • Physical example: damping wings of spectral lines. 1 f (x) 2 (x )2 1 1 x F(x) tan • Wings are so wide that no moments converge! x (x )2 f(x) 0.4 1.4 0.2 1.2 0 1 -4 -2 0 2 4 -0.2 x/ 0.8 F(x) -0.4 0.6 -0.6 0.4 -0.8 0.2 -1 0 -4 -2 0 2 4 x/ Poisson distribution P() 14 6 =5 4 2 19 17 15 13 11 9 7 0 5 • = mean count rate Bin number f (x) e 0.4 x (x i) x! i0 0.3 P(x n) e 0.2 0.1 11 7 9 x 5 3 0 1 P 8 1 – Raindrops in bucket per time interval – Cars on road per time interval – Photons per pixel during exposure 10 3 • A discrete distribution • Describes counting statistics: Counts per bin 12 8 S3 4 2 S1 1 n n! n 0,1, 2,... x 2 (x) (x) x Exponential distribution • Distribution of time intervals between events – Raindrops, cars, photons etc f( x) • A continuous distribution 1 0.5 0 0 1 2 3 2 3 x/ 1 f (x) e 1 x/ F(x) 1 e F(x) 0.5 0 x / 0 x mean time between events 1 x/