Everything about Probability Theory totally explained
Probability theory is the branch of
mathematics concerned with analysis of
random phenomena. The central objects of probability theory are
random variables,
stochastic processes, and
events: mathematical abstractions of
non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. Although an individual coin toss or the roll of a die is a random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted. Two representative mathematical results describing such patterns 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 large sets of data. Methods of probability theory also apply to description of complex systems given only partial knowledge of their state, as in
statistical mechanics. A great discovery of twentieth century
physics was the probabilistic nature of physical phenomena at atomic scales, described in
quantum mechanics.
History
The mathematical theory of
probability has its roots in attempts to analyse
games of chance by
Gerolamo Cardano in the sixteenth century, and by
Pierre de Fermat and
Blaise Pascal in the seventeenth century (for example the "
problem of points").
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. This culminated in modern probability theory, the foundations of which 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. Fairly quickly this became the undisputed
axiomatic basis for modern probability theory.
Treatment
Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory based treatment of probability covers both the discrete, the continuous, any mix of these two and more.
Discrete probability distributions
Discrete probability theory deals with events that occur in
countable sample spaces.
Examples: Throwing
dice, experiments with
decks of cards, and
random walk.
Classical definition:
Initially the probability of an event to occur was defined as number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space.
For example, if the event is "occurrence of an even number when a
die is rolled", the probability is given by
converges in distribution to a
standard normal random variable.
Further Information
Get more info on 'Probability Theory'.
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