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                                     Mean

In probability and statistics, mean and expected value are used synonymously to refer to one measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. In the case of adiscrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability P(x), and then adding all these products together, giving ${\displaystyle \mu =\sum xP(x)}$. An analogous formula applies to the case of acontinuous probability distribution. Not every probability distribution has a defined mean; see the Cauchy distribution for an example. Moreover, for some distributions the mean is infinite: for example, when the probability of the value ${\displaystyle 2^{n}}$ is ${\displaystyle {\tfrac {1}{2^{n}}}}$ for n = 1, 2, 3, ....

Arithmetic mean (AM)[edit]

Main article: Arithmetic mean

The arithmetic mean (or simply "mean") of a sample ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$, usually denoted by ${\displaystyle {\bar {x}}}$, is the sum of the sampled values divided by the number of items in the sample:

${\displaystyle {\bar {x}}={\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}}$

For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is

${\displaystyle {\frac {4+36+45+50+75}{5}}={\frac {210}{5}}=42.}$

Geometric mean (GM)[edit]

The geometric mean is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean) e.g. rates of growth.

${\displaystyle {\bar {x}}=\left(\prod _{i=1}^{n}{x_{i}}\right)^{\tfrac {1}{n}}}$

For example, the geometric mean of five values: 4, 36, 45, 50, 75 is:

${\displaystyle (4\times 36\times 45\times 50\times 75)^{^{1}/_{5}}={\sqrt[{5}]{24\;300\;000}}=30.}$

Harmonic mean (HM)[edit]

The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for examplespeed (distance per unit of time).

${\displaystyle {\bar {x}}=n\cdot \left(\sum _{i=1}^{n}{\frac {1}{x_{i}}}\right)^{-1}}$

For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is

${\displaystyle {\frac {5}{{\tfrac {1}{4}}+{\tfrac {1}{36}}+{\tfrac {1}{45}}+{\tfrac {1}{50}}+{\tfrac {1}{75}}}}={\frac {5}{\;{\tfrac {1}{3}}\;}}=15.}$

                                         

Probability

   Probability is the measure of the likelihood that an event will occur.Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty). The higher the probability of an event, the more certain that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is unbiased, the two outcomes ("head" and "tail") are both equally probable; the probability of "head" equals the probability of "tail". Since no other outcomes are possible, the probability is 1/2 (or 50%), of either "head" or "tail". In other words, the probability of "head" is 1 out of 2 outcomes and the probability of "tail" is also 1 out of 2 outcomes, expressed as 0.5 when converted to decimal, with the above-mentioned quantification system. This type of probability is also called a priori probability.

These concepts have been given an axiomatic mathematical formalization inprobability theory, which is used widely in such areas of study as mathematics,statistics, finance, gambling, science (in particular physics), artificial intelligence/machine learning, computer science, game theory, and philosophyto, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.

Applications[edit]

Probability theory is applied in everyday life in risk assessment and modeling. The insurance industry and markets useactuarial science to determine pricing and make trading decisions. Governments apply probabilistic methods inenvironmental regulation, entitlement analysis (Reliability theory of aging and longevity), and financial regulation.

A good example of the use of probability theory in equity trading is the effect of the perceived probability of any widespread Middle East conflict on oil prices, which have ripple effects in the economy as a whole. An assessment by a commodity trader that a war is more likely can send that commodity's prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are neither assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict.^[22]

In addition to financial assessment, probability can be used to analyze trends in biology (e.g. disease spread) as well as ecology (e.g. biological Punnett squares). As with finance, risk assessment can be used as a statistical tool to calculate the likelihood of undesirable events occurring and can assist with implementing protocols to avoid encountering such circumstances. Probability is used to design games of chance so that casinos can make a guaranteed profit, yet provide payouts to players that are frequent enough to encourage continued play.^[23]

The discovery of rigorous methods to assess and combine probability assessments has changed society.^{[citation needed]} It is important for most citizens to understand how probability assessments are made, and how they contribute to decisions.^{[citation needed]}

Another significant application of probability theory in everyday life is reliability. Many consumer products, such asautomobiles and consumer electronics, use reliability theory in product design to reduce the probability of failure. Failure probability may influence a manufacturer's decisions on a product's warranty.

The cache language model and other statistical language models that are used in natural language processing are also examples of applications of probability theory.