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Moments of probability distributions

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. <x>.

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Moments of probability distributions

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  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 <x> • Mean value (centre of gravity) • Median value (50th percentile) f(x) x F(x) 1 1/2 0 x xmed

  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  +

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

  6. (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/

  7. Counts per bin  = 5 Bin number P 8 4 2 1 Poisson distribution P() • 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  x

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

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