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| | Postcard marketing is a good method to generate traffic to your internet site. It is the simplest way to generate more prospects for your company. It is considered by many entrepreneurs being an old fashioned type of marketing. Postcards are effective and affordable in comparison with other forms of marketing. Postcard provides an excellent exposure to your message since they are usually prepared to read.<br><br>Do not lose out on [http://www.advertisingservicesguide.com/ Advertising Agencies] with promotional items. It is possible to get numerous products customized with whatever data that you like. I enjoy the decorative umbrellas that may be made out of a wide variety of images. If you would like to take people into your new insurance carrier then let them have an umbrella with each new coverage obtained. When you yourself have a couple of available and installed somehow before your company, they definitely catch peoples attention.<br><br><br><br>It is not just a sales letter but an impressive sales letter, implying that I will produce a "kick the doors down" sales letter instead of just a common everyday sales letter. This makes him desire to read more, and immediately brings a confident feeling to the reader. This is a selling ad for a sales letter nevertheless the same rules might be put on any ad that is selling an item.<br><br>Applications like google adsense enable you to make by both ticks or page-impressions. Ticks refer to the clicks of the visitor of one's site for the ads you've inserted. The percentage of clicks per perception is determined by the click-through-rate or CTR. You're paid per-click towards the advertisements on your side. The amount varies is dependent upon lots of aspects. That means it is never regular.<br><br>The procedure is simplest, but requires a lot of considering and approach. That is why it is so complex yet to simple. The entire approach is packed in to a hyperlink. Yes! As amazed as you may be and bewildered, it is actually flagrantly legitimate! All Of The strategists and intellectuals do is place it in the field you see each time you perform a search, package it properly and create the web link.<br><br>You'll find posting mail and service companies that have their own bulk mail permit and could pack your pieces for you. Try your neighborhood yellow pages under "mailing or delivery companies." See below for more information on volume mailing solutions.<br><br>Microsoft assures you a world of unlimited options. It promises to get you wherever you desire to proceed. In line with the firm's course, I'd say that motto has served it well. |
| {{Probability fundamentals}}
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| '''Probability theory''' is the branch of [[mathematics]] concerned with [[probability]], the analysis of [[Statistical randomness|random]] phenomena.<ref>{{cite web|url=http://www.britannica.com/ebc/article-9375936 |title=Probability theory, Encyclopaedia Britannica |publisher=Britannica.com |date= |accessdate=2012-02-12}}</ref> The central objects of probability theory are [[random variable]]s, [[stochastic process]]es, and [[event (probability theory)|event]]s: mathematical abstractions of [[determinism|non-deterministic]] events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. If an individual coin toss or the roll of [[dice]] is considered to be a random event, then if repeated many times the sequence of random events will exhibit certain 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]].
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| 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 descriptions 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]].
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| ==History==
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| The mathematical theory of [[probability]] has its roots in attempts to analyze [[game of chance|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]]"). [[Christiaan Huygens]] published a book on the subject in 1657<ref>{{cite book|last=Grinstead|first=Charles Miller |coauthors=James Laurie Snell|title=Introduction to Probability|pages=vii|chapter=Introduction}}</ref> and in the 19th century a big work was done by [[Laplace]] in what can be considered today as the classic interpretation.<ref>{{cite web|last=Hájek|first=Alan|title=Interpretations of Probability|booktitle=The Stanford Encyclopedia of Philosophy|url=http://plato.stanford.edu/archives/sum2012/entries/probability-interpret/|accessdate=2012-06-20}}</ref>
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| Initially, probability theory mainly considered '''discrete''' events, and its methods were mainly [[combinatorics|combinatorial]]. Eventually, [[mathematical analysis|analytical]] considerations compelled the incorporation of '''continuous''' variables into the theory.
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| 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 [[Kolmogorov axioms|axiom system]] for probability theory in 1933. Fairly quickly this became the mostly undisputed [[axiom system|axiomatic basis]] for modern probability theory but alternatives exist, in particular the adoption of finite rather than countable additivity by [[Bruno de Finetti]].<ref>{{cite web|url=http://www.probabilityandfinance.com/articles/04.pdf |title="The origins and legacy of Kolmogorov's Grundbegriffe", by Glenn Shafer and Vladimir Vovk |format=PDF |date= |accessdate=2012-02-12}}</ref>
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| ==Treatment==
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| 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.
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| ===Motivation===
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| 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 {1,3,5} is an element of the power set of the sample space of die rolls. These collections are called ''events''. In this case, {1,3,5} 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.
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| Probability is a [[Function (mathematics)|way of assigning]] every "event" a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6}) 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 (events that contain no common results, e.g., the events {1,6}, {3}, and {2,4} are all mutually exclusive), the probability that one of the events will occur is given by the sum of the probabilities of the individual events.<ref>Ross, Sheldon. A First course in Probability, 8th Edition. Page 26-27.</ref>
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| The probability that any one of the events {1,6}, {3}, or {2,4} will occur is 5/6. This is the same as saying that the probability of event {1,2,3,4,6} is 5/6. This event encompasses the possibility of any number except five being rolled. The mutually exclusive event {5} has a probability of 1/6, and the event {1,2,3,4,5,6} has a probability of 1, that is, absolute certainty.
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| ===Discrete probability distributions===
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| {{Main|Discrete probability distribution}}
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| '''Discrete probability theory''' deals with events that occur in [[countable]] sample spaces.
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| Examples: Throwing [[dice]], experiments with [[deck of cards|decks of cards]],[[random walk]],and tossing [[coin]]s
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| '''Classical definition:'''
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| 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: see [[Classical definition of probability]].
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| For example, if the event is "occurrence of an even number when a die is rolled", the probability is given by <math>\tfrac{3}{6}=\tfrac{1}{2}</math>, since 3 faces out of the 6 have even numbers and each face has the same probability of appearing.
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| '''Modern definition:'''
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| The modern definition starts with a [[Countable set|finite or countable set]] called the '''[[sample space]]''', which relates to the set of all ''possible outcomes'' in classical sense, denoted by <math>\Omega</math>. It is then assumed that for each element <math>x \in \Omega\,</math>, an intrinsic "probability" value <math>f(x)\,</math> is attached, which satisfies the following properties:
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| #<math>f(x)\in[0,1]\mbox{ for all }x\in \Omega\,;</math>
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| #<math>\sum_{x\in \Omega} f(x) = 1\,.</math>
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| That is, the probability function ''f''(''x'') lies between zero and one for every value of ''x'' in the sample space ''Ω'', and the sum of ''f''(''x'') over all values ''x'' in the sample space ''Ω'' is equal to 1. An '''[[Event (probability theory)|event]]''' is defined as any [[subset]] <math>E\,</math> of the sample space <math>\Omega\,</math>. The '''probability''' of the event <math>E\,</math> is defined as
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| :<math>P(E)=\sum_{x\in E} f(x)\,.</math>
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| So, the probability of the entire sample space is 1, and the probability of the null event is 0.
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| The function <math>f(x)\,</math> mapping a point in the sample space to the "probability" value is called a '''[[probability mass function]]''' abbreviated as '''pmf'''. The modern definition does not try to answer how probability mass functions are obtained; instead it builds a theory that assumes their existence.
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| ===Continuous probability distributions===
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| {{Main|Continuous probability distribution}}
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| '''Continuous probability theory''' deals with events that occur in a continuous sample space.
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| '''Classical definition:'''
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| The classical definition breaks down when confronted with the continuous case. See [[Bertrand's paradox (probability)|Bertrand's paradox]].
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| '''Modern definition:'''
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| If the outcome space of a random variable ''X'' is the set of [[real numbers]] (<math>\mathbb{R}</math>) or a subset thereof, then a function called the '''[[cumulative distribution function]]''' (or '''cdf''') <math>F\,</math> exists, defined by <math>F(x) = P(X\le x) \,</math>. That is, ''F''(''x'') returns the probability that ''X'' will be less than or equal to ''x''.
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| The cdf necessarily satisfies the following properties.
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| #<math>F\,</math> is a [[Monotonic function|monotonically non-decreasing]], [[right-continuous]] function;
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| #<math>\lim_{x\rightarrow -\infty} F(x)=0\,;</math>
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| #<math>\lim_{x\rightarrow \infty} F(x)=1\,.</math>
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| If <math>F\,</math> is [[absolutely continuous]], i.e., its derivative exists and integrating the derivative gives us the cdf back again, then the random variable ''X'' is said to have a '''[[probability density function]]''' or '''pdf''' or simply '''density''' <math>f(x)=\frac{dF(x)}{dx}\,.</math>
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| For a set <math>E \subseteq \mathbb{R}</math>, the probability of the random variable ''X'' being in <math>E\,</math> is
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| :<math>P(X\in E) = \int_{x\in E} dF(x)\,.</math>
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| In case the probability density function exists, this can be written as
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| :<math>P(X\in E) = \int_{x\in E} f(x)\,dx\,.</math>
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| Whereas the ''pdf'' exists only for continuous random variables, the ''cdf'' exists for all random variables (including discrete random variables) that take values in <math>\mathbb{R}\,.</math>
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| These concepts can be generalized for [[Dimension|multidimensional]] cases on <math>\mathbb{R}^n</math> and other continuous sample spaces.
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| ===Measure-theoretic probability theory===
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| The ''raison d'être'' 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.
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| 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 <math>(\delta[x] + \varphi(x))/2</math>, where <math>\delta[x]</math> is the [[Dirac delta function]].
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| 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]]:
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| Given any set <math>\Omega\,</math>, (also called '''sample space''') and a [[sigma-algebra|σ-algebra]] <math>\mathcal{F}\,</math> on it, a [[measure (mathematics)|measure]] <math>P\,</math> defined on <math>\mathcal{F}\,</math> is called a '''probability measure''' if <math>P(\Omega)=1.\,</math>
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| If <math>\mathcal{F}\,</math> is the [[Borel algebra|Borel σ-algebra]] on the set of real numbers, then there is a unique probability measure on <math>\mathcal{F}\,</math> for any cdf, and vice versa. The measure corresponding to a cdf is said to be '''induced''' 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.
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| The ''probability'' of a set <math>E\,</math> in the σ-algebra <math>\mathcal{F}\,</math> is defined as
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| <!--the correct formulation; X has nothing to do with it-->
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| :<math>P(E) = \int_{\omega\in E} \mu_F(d\omega)\,</math>
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| where the integration is with respect to the measure <math>\mu_F\,</math> induced by <math>F\,.</math>
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| Along with providing better understanding and unification of discrete and continuous probabilities, measure-theoretic treatment also allows us to work on probabilities outside <math>\mathbb{R}^n</math>, as in the theory of [[stochastic process]]es. For example to study [[Brownian motion]], probability is defined on a space of functions.
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| ==Probability distributions==
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| {{Main|Probability distributions}}
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| 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 [[uniform distribution (discrete)|discrete uniform]], [[Bernoulli distribution|Bernoulli]], [[binomial distribution|binomial]], [[negative binomial distribution|negative binomial]], [[Poisson distribution|Poisson]] and [[geometric distribution]]s. Important ''continuous distributions'' include the [[uniform distribution (continuous)|continuous uniform]], [[normal distribution|normal]], [[exponential distribution|exponential]], [[gamma distribution|gamma]] and [[beta distribution]]s.
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| ==Convergence of random variables==
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| {{Main|Convergence of random variables}}
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| In probability theory, there are several notions of convergence for [[random variable]]s. They are listed below in the order of strength, i.e., any subsequent notion of convergence in the list implies convergence according to all of the preceding notions.
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| :'''Weak convergence: ''' A sequence of random variables <math>X_1,X_2,\dots,\,</math> converges '''weakly''' to the random variable <math>X\,</math> if their respective cumulative ''distribution functions'' <math>F_1,F_2,\dots\,</math> converge to the cumulative distribution function <math>F\,</math> of <math>X\,</math>, wherever <math>F\,</math> is [[continuous function|continuous]]. Weak convergence is also called '''convergence in distribution'''.
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| ::''Most common shorthand notation:'' <math>X_n \, \xrightarrow{\mathcal D} \, X\,.</math>
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| :'''Convergence in probability:''' The sequence of random variables <math>X_1,X_2,\dots\,</math> is said to converge towards the random variable <math>X\,</math> '''in probability''' if <math>\lim_{n\rightarrow\infty}P\left(\left|X_n-X\right|\geq\varepsilon\right)=0</math> for every ε > 0.
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| ::''Most common shorthand notation:'' <math>X_n \, \xrightarrow{P} \, X\,.</math>
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| :'''Strong convergence:''' The sequence of random variables <math>X_1,X_2,\dots\,</math> is said to converge towards the random variable <math>X\,</math> '''strongly''' if <math>P(\lim_{n\rightarrow\infty} X_n=X)=1</math>. Strong convergence is also known as '''almost sure convergence'''.
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| ::''Most common shorthand notation:'' <math>X_n \, \xrightarrow{\mathrm{a.s.}} \, X\,.</math>
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| As the names indicate, weak convergence is weaker than strong convergence. In fact, strong convergence implies convergence in probability, and convergence in probability implies weak convergence. The reverse statements are not always true.
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| ==Law of large numbers==
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| {{Main|Law of large numbers}}
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| Common intuition suggests that if a fair coin is tossed many times, then ''roughly'' half of the time it will turn up ''heads'', and the other half it will turn up ''tails''. Furthermore, the more often the coin is tossed, the more likely it should be that the ratio of the number of ''heads'' to the number of ''tails'' will approach unity. Modern probability provides a formal version of this intuitive idea, known as the '''law of large numbers'''. This law is remarkable because it is not assumed in the foundations of probability theory, but instead emerges from these foundations as a theorem. Since it links theoretically derived probabilities to their actual frequency of occurrence in the real world, the law of large numbers is considered as a pillar in the history of statistical theory and has had widespread influence.<ref>{{cite web|url=http://www.leithner.com.au/circulars/circular17.htm |title=Leithner & Co Pty Ltd - Value Investing, Risk and Risk Management - Part I |publisher=Leithner.com.au |date=2000-09-15 |accessdate=2012-02-12}}</ref>
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| <!-- Note to editors: Please provide better citation for the historical importance of LLN if you have it -->
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| The '''law of large numbers''' (LLN) states that the sample average | |
| :<math>\overline{X}_n=\frac1n{\sum_{k=1}^n X_k}</math>
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| of a sequence of independent and
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| identically distributed random variables <math>X_k</math> converges towards their common expectation <math>\mu</math>, provided that the expectation of <math>|X_k|</math> is finite.
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| It is in the different forms of [[convergence of random variables]] that separates the ''weak'' and the ''strong'' law of large numbers
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| :<math>
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| \begin{array}{lll}
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| \text{Weak law:} & \overline{X}_n \, \xrightarrow{P} \, \mu & \text{for } n \to \infty \\
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| \text{Strong law:} & \overline{X}_n \, \xrightarrow{\mathrm{a.\,s.}} \, \mu & \text{for } n \to \infty .
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| \end{array}
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| </math>
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| It follows from the LLN that if an event of probability ''p'' is observed repeatedly during independent experiments, the ratio of the observed frequency of that event to the total number of repetitions converges towards ''p''.
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| For example, if <math>Y_1,Y_2,...\,</math> are independent [[Bernoulli distribution|Bernoulli random variables]] taking values 1 with probability ''p'' and 0 with probability 1-''p'', then <math>\textrm{E}(Y_i)=p</math> for all ''i'', so that <math>\bar Y_n</math> converges to ''p'' [[almost surely]].
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| ==Central limit theorem==
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| {{Main|Central limit theorem}}
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| "The central limit theorem (CLT) is one of the great results of mathematics." (Chapter 18 in <ref>[[David Williams (mathematician)|David Williams]], "Probability with martingales", Cambridge 1991/2008</ref>) | |
| It explains the ubiquitous occurrence of the [[normal distribution]] in nature.
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| The theorem states that the [[average]] of many independent and identically distributed random variables with finite variance tends towards a normal distribution ''irrespective'' of the distribution followed by the original random variables. Formally, let <math>X_1,X_2,\dots\,</math> be independent random variables with [[mean]] <math>\mu_\,</math> and [[variance]] <math>\sigma^2 > 0.\,</math> Then the sequence of random variables
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| :<math>Z_n=\frac{\sum_{i=1}^n (X_i - \mu)}{\sigma\sqrt{n}}\,</math>
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| converges in distribution to a [[standard normal]] random variable.
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| ==See also==
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| {{multicol}}
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| *[[Expected value]] and [[Variance]]
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| *[[Fuzzy logic]] and [[Fuzzy measure theory]]
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| *[[Glossary of probability and statistics]]
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| *[[Likelihood function]]
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| *[[List of probability topics]]
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| *[[Catalog of articles in probability theory]]
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| *[[List of publications in statistics]]
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| *[[List of statistical topics]]
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| *[[Probabilistic proofs of non-probabilistic theorems]]
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| {{multicol-break}}
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| *[[Notation in probability]]
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| *[[Predictive modelling]]
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| *[[Probabilistic logic]] – A combination of probability theory and logic
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| *[[Probability axioms]]
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| *[[Probability interpretations]]
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| *[[Statistical independence]]
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| *[[Subjective logic]]
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| {{multicol-end}}
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| ==Notes==
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| {{reflist}}
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| ==References==
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| {{More footnotes|date=September 2009}}
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| *{{cite book
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| | authorlink = Pierre Simon de Laplace
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| | author = Pierre Simon de Laplace
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| | year = 1812
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| | title = Analytical Theory of Probability}}
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| :: The first major treatise blending calculus with probability theory, originally in French: ''Théorie Analytique des Probabilités''.
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| *{{cite book
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| | author = Andrei Nikolajevich Kolmogorov
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| | year = 1950
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| | title = Foundations of the Theory of Probability}}
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| :: The modern measure-theoretic foundation of probability theory; the original German version (''Grundbegriffe der Wahrscheinlichkeitrechnung'') appeared in 1933.
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| *{{cite book
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| | author = [[Patrick Billingsley]]
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| | title = Probability and Measure
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| | publisher = John Wiley and Sons
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| | location = New York, Toronto, London
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| | year = 1979}}
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| * [[Olav Kallenberg]]; ''Foundations of Modern Probability,'' 2nd ed. Springer Series in Statistics. (2002). 650 pp. ISBN 0-387-95313-2
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| *{{cite book
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| | author = [[Henk Tijms]]
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| | year = 2004
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| | publisher = Cambridge Univ. Press
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| | title = Understanding Probability}}
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| :: A lively introduction to probability theory for the beginner.
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| * Olav Kallenberg; ''Probabilistic Symmetries and Invariance Principles''. Springer -Verlag, New York (2005). 510 pp. ISBN 0-387-25115-4
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| *{{cite book
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| | last = Gut
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| | first = Allan
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| | title = Probability: A Graduate Course
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| | publisher = Springer-Verlag
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| | year = 2005
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| | isbn = 0-387-22833-0
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| }}
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| ==External links==
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| * [http://www.youtube.com/watch?v=9eaOxgT5ys0 Animation] on the probability space of dice.
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| {{Mathematics-footer}}
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| {{DEFAULTSORT:Probability Theory}}
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| [[Category:Probability theory| ]]
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| {{Link FA|ka}}
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| [[id:Peluang (matematika)]]
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