1 / 11

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>.

bryson
Download Presentation

Moments of probability distributions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related