Hey-Let’s talk about distributions!

What do we mean by distribution?

OBSERVE: the number of successes in \(N\) tosses of a coin.

Say head is a success (1), tail a failure (0).

Support of distribution is the integer interval \({0,1,\ldots, N}\)

We write \(X \sim Binom(p,N)\) to describe the random number of successes \(X\) obtained in \(N\) independent tosses of a coin which has probability \(p\) of producing HEAD (a success).

Equivalently you can flip \(N\) identical coins.

Each sample gives an outcomes that is a single number e.g. \(X=3\).

Let us tabulate and count all the mannners in which a given outcome can occur, in a series of \(N=4\) coin flips.

0000 0 1 way to get 0

1000 1 4 ways to get 1

0100 1

0010 1

0001 1

1100 2 6 ways to get 2

1010 2

1001 2

0110 2

0101 2

0011 2

1110 3 4 ways to get 3

0111 3

1011 3

1101 3

1111 4 1 way to get 4

\((1+x)^4 = (1+2x+x^2)(1+2x+x^2) = 1+2x+x^2+2x+4x^2+x^3 +x^2+2x^3+x^4\)

This is:

\(1 + 4x + 6x2 + 4x^3 + x4\)

Look at the coefficients !

OBSERVE: the duration of inter-events (quiet) periods of the Old Faithful Geyser (N = 222 episods 1978-1979)

This Geyser is located in the Yellowstone National Park

temptative statistical model

Physical explanation

Better yet and admission of ignorance

OBSERVE: the pixel value of bw photographs

LOW_KEY: Ansel Adams

HIGH-KEY: Misha Gregory Macaw

OBSERVE: mutation events in genes

MUTATIONS of genes and their frequencies:

This is a sorted histogram

Fluorescence in Situ Hybridization (detects chromosomal rearrangements events)

OBSERVE: HEIGHT of HUMAN POPULATIONS

OBSERVE: size and body mass of animals

ABUNDANCE OF Sockeye salmon BY SIZE: (multimodal!)

OBSERVE: INCIDENCE OF ADVERSE EVENT (RISKS/DISEASE/DEATH)

Lamb survival

ACUTE APPENDICITIS (bimodal age incidence):

RISK OF DYING AFTER AN EMERGENCY (TRAUMA DEATH) (trimodal)

OBSERVE: individual atom disintegration events

GEIGER COUNTS them with this apparatus, which triggers an avalanche of electrons in a gaz chamber at each emission of a single charged particle (alpha) by the disintegrating atom

FIRE ALARM: SMOKE DETECTOR based on the Geiger counter:

Say (this is a fictitious example) that the number of false fire alarms in a Halifax averages 2.1 per day.

What is the chance that we might have a day with 4 false alarms? 17 false alarms? 1 false alarm?

How close are real distributions from model distributions

It depends on the data.

But bear in mind that sometimes (i.e. for certain sets of measurements) the agreement can be close!

OBSERVE: DEATH by Horse kicks in the Prussian Cavalry

SOME DISTRIBUTIONS FOUND IN PHYSICS

Kinetic theory of gases

OBSERVE: Velocities of individual molecules in a gas

(Maxwell velocity distribution)

Nuclear particles

OBSERVE: The lifetimes of muons

(Histogram of lifetime. Qualitatively Poisson-like) src: https://www.phys.ufl.edu/courses/phy4803L/group_I/gamma_gamma/poisson.pdf

Astronomy

OBSERVE: The motion of stars in the center of our galaxy

The kinematics of large and dense groups of stars is often approximated by a Maxwellian velocity distribution (Maxwell–Boltzmann distribution with a Gaussian distribution of velocities). The reason is that this distribution solves the collisionless Boltzmann equations of stellar dynamics.

See Stellar Dynamics

Stars clusters, globular clusters, and even clusters of galaxies are modeled as systems of particles that approximately obey a Maxwell distribution.

Several physical effects (tides, evaporations etc.) will lead to departures of stellar velocities from the Maxwell distribution.

Radioactive decay of atoms

OBSERVE: individual decaying atoms (radioactive disintegration)

Experimental setup

Cobalt 60 (more normal than poissonian): src:https://www.hiram.edu/wp-content/uploads/2018/04/GeigerTube_S18-1.pdf

Radioactive materials disintegrate in a completely random manner. There exists for any radioactive substance a certain probability that any particular nucleus will emit radiation within a given time interval.

This probability is the same for all nuclei of the same type and is characteristic of that type of nucleus. There is no way to predict the time at which an individual nucleus will decay.

However, when a large number of disintegrations take place, there is a definite average decay rate which is characteristic of the particular nuclear type.

OBSERVE: Counts from a Po-210 source (Rutherford original data)

Decay of Polonium into lead

Polonium

Rutherford, Geiger, and Bateman, ‘The Probability Variations in the Distribution of alpha Particles’, Phil. Mag. 20, 698, (1910).

src: http://home.sandiego.edu/~severn/p272/nuclear-counting-statistics-sp18.html

Fit by a Poisson distribution:

Strength of materials

Weibull distribution

The smaller the diameter, the stronger is the fiber !

Models the (mechanical, electrical, biological) rupture of systems aa being controlled by the weakest link/element in the system.

src: https://www.sciencedirect.com/science/article/pii/S2238785417302442

A physical examination corroborates the conceptual statistical model of Weibull:

A SEM observation of the tip of representative tensile-ruptured fibers, shown in Fig. 6, provided further evidence of a fracture mechanism that could justify the hyperbolic correlation in Eq. (2). With the same magnification, the thinner fiber, Fig. 6(a), shows a more uniform fracture associated with lesser fibrils. By contrast, a fiber with larger diameters, Fig. 6(b), displays a relatively non-uniform fracture with participation of more fibrils. Statistically, there is always a chance that one of the many fibrils of the thicker giant bamboo fiber in Fig. 6(b), would prematurely break and then act as a flaw to cause the fiber rupture at a lower stress as compared to the thinner fiber, Fig. 6(a). In other words, the larger distribution of fibrils strength of the thicker fiber allows one of them breaking shortly than any of the fewer fibrils of a thinner fiber. Wang and Shao [19] also attributed the reduction in mechanical properties of bamboo fibers to accumulation of defects within fiber diameter variations.

Loi de Weibull (fr)

There is light and light !

Counting photons may give clues as to the mechanism of light emission:

Comparison of the Poisson and Bose-Einstein distributions. The Poisson distribution is characteristic of coherent light while the Bose-Einstein distribution is characteristic of thermal light. Both distribution have the same expectation value

Reason of connection between Poisson and coherent light:

src:https://physics.stackexchange.com/questions/296106/why-do-coherent-states-have-poisson-number-distribution

In quantum mechanics, a coherent state of a quantum harmonic oscillator (QHO) is an eigenstate of the lowering operator. Expanding in the number basis, we find that the number of photons in a coherent state follows a Poisson distribution.

Bose-Einstein statistics (thermal or non-coherent emission) is much more fascinating!

Explains superconductivity of Helium, allows us to create atom lasers and Einstein condensate of atoms

There is only room for one: Fermi-Dirac statistics

In a Fermi gas, no two fermion particles can be in the same quantum state, This implies that particles must obey the exclusion principle and has striking implications on the velocity distribution of particules in the gas. Fermions have a half-integer spin (magnetic moment), Electrons, protons or neutrons are examples of fermions.

src:https://readingfeynman.org/tag/fermi-dirac-statistics/

OBSERVE: velocities of particles in a Fermi-Dirac gas

Fermi-Dirac statistics is the opposite of the Bose-Einstein statistics. The first is individualistic (everybody needs to be isolated in a private bubble), the second is gregarious (can form condensates and reveal quantum coherence at the macroscopic level).

The mathematical derivation (or justification) of the Fermi-Dirac statistics is given e.g. here or there

Neutron stars and pulsars

Stars can become unstable and explode, which gives rise to a supernova (a famous example being the crab Nebulae, formed in 1054 A.D.).

During the explosion, the outer shell expands and the core of the supernova collapse into a very dense object: a neutron star.

The Fermi-Dirac statistics predicts that this star will acquire magnetic properties.

Pulsars are magnetized rotating neutrin stars that emit a powerful beak of electromagnetic radiation in a narrow direction aligned with the magnetic axis of the star (Think of the rotating light beam of a lighthouse).

Here is a plot of the distribution of the rotation periods of 1533 pulsars observed since 1967:

src:http://physik.uni-graz.at/~dk-user/talks/Tolos1_GD1117.pdf

OBSERVE: period of rotation of pulsars

Spectral profiles

Although each atom radiates light at precise energy levels, spectroscopic measures of light emission of a gas (example from a star or from the earth’s atmosphere) shows not a sharp Dirac peak of light emission occurring at a single energy level (or frequency) but a ‘broad peak’ spread on a range of frequency.

The exact shape of absorption or emission profiles is influenced by the physical processes that take place in the medium where the light is emitted.

Several distributions have been proposed to fit these profiles. One of them is due to the physicist (and Nobel Prize) Henrik Lorentz. It coincides in fact with a probability distribution proposed a century earlier by Cauchy.

The Lorentz-Cauchy distribution is so broad that it does not have a finite variance, nor in fact any finite moment of order greater or equal to one. This means that it has no average and no precisely defined ‘width’. This pathological behaviour caused some stir between mathematicians…

OBSERVE: absorption of emission of light

see Atmospheric spectroscopy

SOME DISTRIBUTIONS FOUND IN BIOLOGY

GENOMICS

OBSERVE: GENES SHARED BY SIBLINGS

src: Assumption-Free Estimation of Heritability (Peter M Visscher and al. ,2006)

Empirical Distribution of Actual Additive Genetic Relationships of 4,401 Quasi-Independent Pairs of Full Sibs

Histogram of the genome-wide additive genetic relationships of full-sib pairs estimated from genetic markers:

GENETICS

SOME DISTRIBUTIONS FOUND IN OTHER FIELDS

Hurricanes

Are hurricanes following a Poisson distribution?

OBSERVE: number of hurricanes in a year

Car crashes on small two lanes rural roads

Poisson or negative binomial?

Some observed distributions of counts show more dispersion (higher variance) than predicted by a Poisson distribution. In these cases, the negative binomial can provide a better fit to the data.

OBSERVE: Number of car crashes per road in rural areas

N-gram distributions

Cavnar-Trenckle: application to language identification

OBSERVE: the frequency of adjacent groups of letters in human languages (written texts)

src:http://practicalcryptography.com/cryptanalysis/letter-frequencies-various-languages/

Trigram frequency: english

Trigram frequency: french

Digital humanities: Evolution of antique greek tragedies

src: http://dh.obdurodon.org/R-script.xhtml

OBSERVE: Number of works in antique greek tragedies

HINTS:

why is Old Faithful bimodal?

(src: https://www.knowablemagazine.org/)