**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Poisson distribution

Summary

In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson ('pwɑːsɒn; pwasɔ̃). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume.
It plays an important role for discrete-stable distributions.
For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution with mean 3: the most likely numbers are 2 and 3 but 1 and 4 are also likely and there is a small probability of it being as low as zero and a very small probability it could be 10.
Another example is the number of decay events that occur from a radioactive source during a defined observation period.
The distribution was first introduced by Siméon Denis Poisson (1781–1840) and published together with his probability theory in his work Recherches sur la probabilité des jugements en matière criminelle et en matière civile (1837). The work theorized about the number of wrongful convictions in a given country by focusing on certain random variables N that count, among other things, the number of discrete occurrences (sometimes called "events" or "arrivals") that take place during a time-interval of given length. The result had already been given in 1711 by Abraham de Moivre in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus . This makes it an example of Stigler's law and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre.
In 1860, Simon Newcomb fitted the Poisson distribution to the number of stars found in a unit of space.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (15)

Related MOOCs (15)

Related courses (146)

Related people (11)

Related concepts (104)

Related units (3)

Advanced statistical physics

We explore statistical physics in both classical and open quantum systems. Additionally, we will cover probabilistic data analysis that is extremely useful in many applications.

Advanced statistical physics

We explore statistical physics in both classical and open quantum systems. Additionally, we will cover probabilistic data analysis that is extremely useful in many applications.

Neuronal Dynamics - Computational Neuroscience of Single Neurons

The activity of neurons in the brain and the code used by these neurons is described by mathematical neuron models at different levels of detail.

Les étudiants comprennent les bases physico-chimiques des méthodes de séparation chromatographiques et électrophorétiques.

Introduction to notions of probability and basic statistics.

A basic course in probability and statistics

Geometric distribution

In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: The probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set ; The probability distribution of the number Y = X − 1 of failures before the first success, supported on the set . Which of these is called the geometric distribution is a matter of convention and convenience. These two different geometric distributions should not be confused with each other.

Negative binomial distribution

In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of successes (denoted ) occurs. For example, we can define rolling a 6 on a dice as a success, and rolling any other number as a failure, and ask how many failure rolls will occur before we see the third success ().

Random matrix

In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice. In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms.

In this thesis, we study the 3 challenges described above. First, we study different reconstruction techniques and assess the fidelity of each reconstruction results by means of structured illumination and phase conjugation. By reconstructing the 3D refractive index of the sample using different algorithms (i.e. Born, Rytov, and Radon) and then perform a numerical back-propagation of experimentally measured structured illumination pattern we are able to assess the fidelity of each reconstruction algorithms without prior information about the 3D RI distribution of the sample.The second part of the thesis is concerned with the 3D reconstruction of samples using intensity-only measurements which the need to holographically acquire them. We show that using intensity-only measurements, we could still be able to reconstruct the 3D volume of the sample with edge-enhanced effects which was proven useful for drug delivery applications in which nano-particles were identified on the cell membrane of immune T-cells in a drug delivery studies. Such reconstruction technique would result in more robust imaging system where the commercial imaging microscope systems can be incorporated with LEDs for high-quality speckle noise-free imaging systems. In addition, we show that under certain conditions, we can be able to reconstruct the 3D refractive index distribution of different samples.The third part of the thesis is contributing to high-speed complex wave-front shaping using DMDs. In that part, new modulation technique is demonstrated that can boost the speed of the current time-multiplexing techniques by a factor of 32. The modulation technique is based on amplitude modulation where an amplitude modulator is synchronized withvthe DMD to modulate the intensity of each bit-plane of an 8-bit image and then all the modulated bit-planes are linearly added on the detector. Such modulation technique can be used not only for structured illumination microscopy but also for high-speed 3D printing applications as well as projectors.The last part is concerned with using deep learning approaches to solve the missing cone problem usually accompanied with optical imaging due to the limited numerical aperture of the imaging system. Two techniques are discussed; the first is based on using a physical model to enhance the quality of the 3D RI reconstruction and the second is based on using deep neural network to solve the missing cone problem.

Yinlin Hu, Yunpeng Li, Jingyu Li, Rui Song

Semantic segmentation for remote sensing images (RSI) is critical for the Earth monitoring system. However, the covariate shift between RSI datasets under different capture conditions cannot be alleviated by directly using the unsupervised domain adaptation (UDA) method, which negatively affects the segmentation accuracy in RSI. We propose a stepwise domain adaptive segmentation network with covariate shift alleviation (Cov-DA) for RSI parsing to solve this issue. Specifically, to alleviate domain shift generated by different sensors, both the source and target domains are projected into a colorspace with normalized distribution through an elaborate colorspace mapping unified module (CMUM). The color distributions of these two domains tend to be more uniform. Furthermore, in the target domain, the multistatistics joint evaluation module (MJEM) is proposed to capture different statistical characteristics of subscenarios for selecting plain scenarios regarded as high-confidence segmentation results to assist the further improvement of segmentation performance. In addition, a pyramid perceptual attention module (PPAM) containing omnidirectional features without computational burdens is added to our network for effectively enhancing the multiscale feature capture ability. In the cross-city DA experiments based on the International Society for Photogrammetry and Remote Sensing (ISPRS) and aerial benchmarks, the superiority of our algorithm is significantly demonstrated. Furthermore, we release a large-scale Martian terrain dataset noted as "Mars-Seg" containing 5 K images with pixel-level accurate annotations regarding issues, such as the lack of semantic segmentation datasets for unknown scenes.

We introduce a protocol addressing the conformance test problem, which consists in determining whether a process under test conforms to a reference one. We consider a process to be characterized by the set of end products it produces, which is generated according to a given probability distribution. We formulate the problem in the context of hypothesis testing and consider the specific case in which the objects can be modeled as pure loss channels. We demonstrate theoretically that a simple quantum strategy, using readily available resources and measurement schemes in the form of two-mode squeezed vacuum and photon counting, can outperform any classical strategy. We experimentally implement this protocol, exploiting optical twin beams, validating our theoretical results, and demonstrating that, in this task, there is a quantum advantage in a realistic setting.

Related lectures (999)

Plasma Physics: Collisions and ResistivityPHYS-325: Introduction to plasma physics

Covers Coulomb collisions and resistivity in plasma, highlighting their random walk nature.

Extreme Problems in Diophantine Approximation and Dynamics

Explores the Logarithmic Law, geodesic flows, hyperbolic surfaces, and rare events in probability.

Rainfall Models: Deterministic vs StochasticENV-424: Water resources engineering

Covers deterministic and stochastic rainfall models in water resources engineering, including generation, calibration, and spatially explicit models.