Randomness in the context of "Terrorism"

A disease cluster is an unusually large aggregation of a relatively uncommon disease (medical condition) or event within a particular geographical location or period. Recognition of a cluster depends on its size being greater than would be expected by chance. Identification of a suspected disease cluster may initially depend on anecdotal evidence. Epidemiologists and biostatisticians then assess whether the suspected cluster corresponds to an actual increase of disease in the area. Typically, when clusters are recognized, they are reported to public health departments in the local area. If clusters are of sufficient size and importance, they may be re-evaluated as outbreaks.

John Snow's pioneering investigation of the 1854 cholera outbreak in Soho, London, is seen as a classic example of the study of such a cluster.

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).

For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads, and 0.5 for X = tails (assuming that the coin is fair). More commonly, probability distributions are used to compare the relative occurrence of many different random values.

Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is considered radioactive. Three of the most common types of decay are alpha, beta, and gamma decay. The weak force is the mechanism that is responsible for beta decay, while the other two are governed by the electromagnetic and nuclear forces.

Radioactive decay is a random process at the level of single atoms. According to quantum theory, it is impossible to predict when a particular atom will decay, regardless of how long the atom has existed. However, for a significant number of identical atoms, the overall decay rate can be expressed as a decay constant or as a half-life. The half-lives of radioactive atoms have a huge range: from nearly instantaneous to far longer than the age of the universe.

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical sources.

This motion pattern typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the equipartition theorem).

A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function in which

• the domain is the set of possible outcomes in a sample space (e.g. the set ${\displaystyle \{H,T\}}$ which are the possible upper sides of a flipped coin heads ${\displaystyle H}$ or tails ${\displaystyle T}$ as the result from tossing a coin); and
• the range is a measurable space (e.g. corresponding to the domain above, the range might be the set ${\displaystyle \{-1,1\}}$ if say heads ${\displaystyle H}$ mapped to -1 and ${\displaystyle T}$ mapped to 1). Typically, the range of a random variable is a subset of the real numbers.

Informally, randomness typically represents some fundamental element of chance, such as in the roll of a die; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup.

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause or prevent a tornado in Texas.

Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution and is fully determined by their initial conditions, with no random elements involved. In other words, despite the deterministic nature of these systems, this does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:

In probability theory, an experiment or trial (see below) is the mathematical model of any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. An experiment is said to be random if it has more than one possible outcome, and deterministic if it has only one. A random experiment that has exactly two (mutually exclusive) possible outcomes is known as a Bernoulli trial.

When an experiment is conducted, one (and only one) outcome results— although this outcome may be included in any number of events, all of which would be said to have occurred on that trial. After conducting many trials of the same experiment and pooling the results, an experimenter can begin to assess the empirical probabilities of the various outcomes and events that can occur in the experiment and apply the methods of statistical analysis.