Probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event.
Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes.
Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.
As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics or sequential estimation. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics.
History of probability
The modern mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century. Christiaan Huygens published a book on the subject in 1657. In the 19th century, what is considered the classical definition of probability was completed by Pierre Laplace.Initially, probability theory mainly considered events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of variables into the theory.
This culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933. This became the mostly undisputed axiomatic basis for modern probability theory; but, alternatives exist, such as the adoption of finite rather than countable additivity by Bruno de Finetti.
Treatment
Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, and more.Motivation
Consider an experiment that can produce a number of outcomes. The set of all outcomes is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the power set of the sample space of dice rolls. These collections are called events. In this case, is the event that the die falls on some odd number. If the results that actually occur fall in a given event, that event is said to have occurred.Probability is a way of assigning every "event" a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one. To qualify as a probability distribution, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events, the probability that any of these events occurs is given by the sum of the probabilities of the events.
The probability that any one of the events,, or will occur is 5/6. This is the same as saying that the probability of event is 5/6. This event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, and the event has a probability of 1, that is, absolute certainty.
When doing calculations using the outcomes of an experiment, it is necessary that all those elementary events have a number assigned to them. This is done using a random variable. A random variable is a function that assigns to each elementary event in the sample space a real number. This function is usually denoted by a capital letter. In the case of a die, the assignment of a number to certain elementary events can be done using the identity function. This does not always work. For example, when flipping a coin the two possible outcomes are "heads" and "tails". In this example, the random variable X could assign to the outcome "heads" the number "0" and to the outcome "tails" the number "1".
Discrete probability distributions
deals with events that occur in countable sample spaces.Examples: Throwing dice, experiments with decks of cards, random walk, and tossing coins.
Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see Classical definition of probability.
For example, if the event is "occurrence of an even number when a dice is rolled", the probability is given by, since 3 faces out of the 6 have even numbers and each face has the same probability of appearing.
The modern definition starts with a finite or countable set called the sample space, which relates to the set of all possible outcomes in classical sense, denoted by. It is then assumed that for each element, an intrinsic "probability" value is attached, which satisfies the following properties:
So, the probability of the entire sample space is 1, and the probability of the null event is 0.
The function mapping a point in the sample space to the "probability" value is called a abbreviated as.
Continuous probability distributions
deals with events that occur in a continuous sample space.The classical definition breaks down when confronted with the continuous case. See Bertrand's paradox.
If the sample space of a random variable X is the set of real numbers or a subset thereof, then a function called the exists, defined by. That is, F returns the probability that X will be less than or equal to x.
The CDF necessarily satisfies the following properties.
- is a monotonically non-decreasing, right-continuous function;
For a set, the probability of the random variable X being in is
In case the PDF exists, this can be written as
Whereas the PDF exists only for continuous random variables, the CDF exists for all random variables that take values in
These concepts can be generalized for multidimensional cases on and other continuous sample spaces.
Measure-theoretic probability theory
The utility of the measure-theoretic treatment of probability is that it unifies the discrete and the continuous cases, and makes the difference a question of which measure is used. Furthermore, it covers distributions that are neither discrete nor continuous nor mixtures of the two.An example of such distributions could be a mix of discrete and continuous distributions—for example, a random variable that is 0 with probability 1/2, and takes a random value from a normal distribution with probability 1/2. It can still be studied to some extent by considering it to have a PDF of, where is the Dirac delta function.
Other distributions may not even be a mix, for example, the Cantor distribution has no positive probability for any single point, neither does it have a density. The modern approach to probability theory solves these problems using measure theory to define the probability space:
Given any set and a σ-algebra on it, a measure defined on is called a if
If is the Borel σ-algebra on the set of real numbers, then there is a unique probability measure on for any CDF, and vice versa. The measure corresponding to a CDF is said to be by the CDF. This measure coincides with the pmf for discrete variables and PDF for continuous variables, making the measure-theoretic approach free of fallacies.
The probability of a set in the σ-algebra is defined as
where the integration is with respect to the measure induced by
Along with providing better understanding and unification of discrete and continuous probabilities, measure-theoretic treatment also allows us to work on probabilities outside, as in the theory of stochastic processes. For example, to study Brownian motion, probability is defined on a space of functions.
When it is convenient to work with a dominating measure, the Radon–Nikodym theorem is used to define a density as the Radon–Nikodym derivative of the probability distribution of interest with respect to this dominating measure. Discrete densities are usually defined as this derivative with respect to a counting measure over the set of all possible outcomes. Densities for absolutely continuous distributions are usually defined as this derivative with respect to the Lebesgue measure. If a theorem can be proved in this general setting, it holds for both discrete and continuous distributions as well as others; separate proofs are not required for discrete and continuous distributions.