Probability Theory
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- axiom
- An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting pint for further reasoning and arguments. The word comes from ancient Greek word meaning
that which is thought worthy or fit
orthat which commends itself as evident
- An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting pint for further reasoning and arguments. The word comes from ancient Greek word meaning
- Law of Large Numbers
- The law of large numbers (LLN) is a mathematical law that states that the average of the results obtained form a large number of independent random samples converges to the true value, if it exists. More formally, the LLN states that given a sample of independent and identically distributed values, the sample mean converges to the true mean.
- Central Limit Theorem
- The central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. This holds even if the original variables themselves are not normally distributed. There are several versions of the CLT, each applying in the context of different conditions.
Notes
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability implementations, probability theory treats the concepts in a rigorous mathematical manner by expressing it through a set of axioms. Typically, these axioms formalize 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.
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 behavior 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.
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. In the 19th century, what is considered the classical definition of probability was completed by Pierre Laplace.
Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory. The foundations of modern probability theory were 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 axiomatic basis for modern probability theory.
The set of all outcomes of an experiment is called the sample space of the experiment. Th power set of the sample space (or equivalently, the event space) is formed by considering all collections of possible results. 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 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 all probabilities of the events.
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.
Discrete probability theory deals with events that occur in countable sample spaces. Classical definition: Initially the probability of an event to occur was defines as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space. Modern Definition: 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 . It is then assumed that for each element , an intrinsic "probability" value is attached, which satisfies the following properties:
An event is defined as any subset of the sample space . The probability of the event is defined as:
So, the probability of the entire sample space if 1, and the probability of the null event is 0.
Certain random variables occur very often in probability theory because they well describe many natural or physical processes. Their distributions, therefore, have gained special importance in probability theory. Some fundamental discrete distributions are the discrete uniform, Bernoulli, binomial, negative binomial, Poisson and geometric distributions. Important continuous distributions include the continuous uniform, normal, exponential, gamma and beta distributions.
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