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76-508: The simplest way of measuring the home range is to construct the smallest possible convex polygon around the data but this tends to overestimate the range. The best known methods for constructing utilization distributions are the so-called bivariate Gaussian or normal distribution kernel density methods . More recently, nonparametric methods such as the Burgman and Fox's alpha-hull and Getz and Wilmers local convex hull have been used. Software

152-714: 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. 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

228-469: A measure P {\displaystyle P\,} defined on F {\displaystyle {\mathcal {F}}\,} is called a probability measure if P ( Ω ) = 1. {\displaystyle P(\Omega )=1.\,} If F {\displaystyle {\mathcal {F}}\,} is the Borel σ-algebra on the set of real numbers, then there

304-603: A normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable . The general form of its probability density function is f ( x ) = 1 2 π σ 2 e − ( x − μ ) 2 2 σ 2 . {\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,.} The parameter μ {\textstyle \mu }

380-467: A sequence of independent and identically distributed random variables X k {\displaystyle X_{k}} converges towards their common expectation (expected value) μ {\displaystyle \mu } , provided that the expectation of | X k | {\displaystyle |X_{k}|} is finite. It is in the different forms of convergence of random variables that separates

456-502: 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 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, on foundations laid by Andrey Nikolaevich Kolmogorov . Kolmogorov combined

532-636: A continuous sample space. Classical definition : The classical definition breaks down when confronted with the continuous case. See Bertrand's paradox . Modern definition : If the sample space of a random variable X is the set of real numbers ( R {\displaystyle \mathbb {R} } ) or a subset thereof, then a function called the cumulative distribution function ( CDF ) F {\displaystyle F\,} exists, defined by F ( x ) = P ( X ≤ x ) {\displaystyle F(x)=P(X\leq x)\,} . That is, F ( x ) returns

608-405: A fixed collection of independent normal deviates is a normal deviate. Many results and methods, such as propagation of uncertainty and least squares parameter fitting, can be derived analytically in explicit form when the relevant variables are normally distributed. A normal distribution is sometimes informally called a bell curve . However, many other distributions are bell-shaped (such as

684-758: A generic normal distribution with density f {\textstyle f} , mean μ {\textstyle \mu } and variance σ 2 {\textstyle \sigma ^{2}} , the cumulative distribution function is F ( x ) = Φ ( x − μ σ ) = 1 2 [ 1 + erf ⁡ ( x − μ σ 2 ) ] . {\displaystyle F(x)=\Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]\,.} The complement of

760-546: A known approximate solution, x 0 {\textstyle x_{0}} , to the desired Φ ( x ) {\textstyle \Phi (x)} . x 0 {\textstyle x_{0}} may be a value from a distribution table, or an intelligent estimate followed by a computation of Φ ( x 0 ) {\textstyle \Phi (x_{0})} using any desired means to compute. Use this value of x 0 {\textstyle x_{0}} and

836-508: 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 Ω {\displaystyle \Omega \,} (also called sample space ) and a σ-algebra F {\displaystyle {\mathcal {F}}\,} on it,

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912-628: A purely fixed-point LoCoH method to fixed radius and adaptive point/radius LoCoH methods. Although, currently, more software is available to implement parametric than nonparametric methods (because the latter approach is newer), the cited papers by Getz et al. demonstrate that LoCoH methods generally provide more accurate estimates of home range sizes and have better convergence properties as sample size increases than parametric kernel methods. Home range estimation methods that have been developed since 2005 include: Computer packages for using parametric and nonparametric kernel methods are available online. In

988-629: A random fashion). 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

1064-493: 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 ( δ [ x ] + φ ( x ) ) / 2 {\displaystyle (\delta [x]+\varphi (x))/2} , where δ [ x ] {\displaystyle \delta [x]} is the Dirac delta function . Other distributions may not even be

1140-429: 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 (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in

1216-542: A variance of ⁠ 1 2 {\displaystyle {\frac {1}{2}}} ⁠ , and Stephen Stigler once defined the standard normal as φ ( z ) = e − π z 2 , {\displaystyle \varphi (z)=e^{-\pi z^{2}},} which has a simple functional form and a variance of σ 2 = 1 2 π . {\textstyle \sigma ^{2}={\frac {1}{2\pi }}.} Every normal distribution

1292-424: Is a normal deviate with parameters μ {\textstyle \mu } and σ 2 {\textstyle \sigma ^{2}} , then this X {\textstyle X} distribution can be re-scaled and shifted via the formula Z = ( X − μ ) / σ {\textstyle Z=(X-\mu )/\sigma } to convert it to

1368-488: Is a unique probability measure on F {\displaystyle {\mathcal {F}}\,} 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. The probability of a set E {\displaystyle E\,} in

1444-730: Is a version of the standard normal distribution, whose domain has been stretched by a factor σ {\textstyle \sigma } (the standard deviation) and then translated by μ {\textstyle \mu } (the mean value): f ( x ∣ μ , σ 2 ) = 1 σ φ ( x − μ σ ) . {\displaystyle f(x\mid \mu ,\sigma ^{2})={\frac {1}{\sigma }}\varphi \left({\frac {x-\mu }{\sigma }}\right)\,.} The probability density must be scaled by 1 / σ {\textstyle 1/\sigma } so that

1520-778: Is advantageous because of a much simpler and easier-to-remember formula, and simple approximate formulas for the quantiles of the distribution. Normal distributions form an exponential family with natural parameters θ 1 = μ σ 2 {\textstyle \textstyle \theta _{1}={\frac {\mu }{\sigma ^{2}}}} and θ 2 = − 1 2 σ 2 {\textstyle \textstyle \theta _{2}={\frac {-1}{2\sigma ^{2}}}} , and natural statistics x and x . The dual expectation parameters for normal distribution are η 1 = μ and η 2 = μ + σ . The cumulative distribution function (CDF) of

1596-394: Is also used quite often. The normal distribution is often referred to as N ( μ , σ 2 ) {\textstyle N(\mu ,\sigma ^{2})} or N ( μ , σ 2 ) {\textstyle {\mathcal {N}}(\mu ,\sigma ^{2})} . Thus when a random variable X {\textstyle X}

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1672-456: Is attached, which satisfies the following properties: 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 is defined as any subset E {\displaystyle E\,} of the sample space Ω {\displaystyle \Omega \,} . The probability of

1748-410: Is available for using both parametric and nonparametric kernel methods. The concept of the home range can be traced back to a publication in 1943 by W. H. Burt, who constructed maps delineating the spatial extent or outside boundary of an animal's movement during the course of its everyday activities. Associated with the concept of a home range is the concept of a utilization distribution , which takes

1824-865: Is below a chosen acceptably small error, such as 10 , 10 , etc.: x n + 1 = x n − Φ ( x n , x 0 , Φ ( x 0 ) ) − Φ ( desired ) Φ ′ ( x n ) , {\displaystyle x_{n+1}=x_{n}-{\frac {\Phi (x_{n},x_{0},\Phi (x_{0}))-\Phi ({\text{desired}})}{\Phi '(x_{n})}}\,,} where Φ ′ ( x n ) = 1 2 π e − x n 2 / 2 . {\displaystyle \Phi '(x_{n})={\frac {1}{\sqrt {2\pi }}}e^{-x_{n}^{2}/2}\,.} Probability theory Probability theory or probability calculus

1900-417: Is called a normal deviate . Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem . It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance

1976-838: Is described by this probability density function (or density): φ ( z ) = e − z 2 2 2 π . {\displaystyle \varphi (z)={\frac {e^{\frac {-z^{2}}{2}}}{\sqrt {2\pi }}}\,.} The variable z {\textstyle z} has a mean of 0 and a variance and standard deviation of 1. The density φ ( z ) {\textstyle \varphi (z)} has its peak 1 2 π {\textstyle {\frac {1}{\sqrt {2\pi }}}} at z = 0 {\textstyle z=0} and inflection points at z = + 1 {\textstyle z=+1} and z = − 1 {\textstyle z=-1} . Although

2052-412: Is equivalent to saying that the standard normal distribution Z {\textstyle Z} can be scaled/stretched by a factor of σ {\textstyle \sigma } and shifted by μ {\textstyle \mu } to yield a different normal distribution, called X {\textstyle X} . Conversely, if X {\textstyle X}

2128-469: Is given by the sum of the probabilities of the events. 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. When doing calculations using

2204-437: Is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as measurement errors , often have distributions that are nearly normal. Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any linear combination of

2280-457: Is normally distributed with mean μ {\textstyle \mu } and standard deviation σ {\textstyle \sigma } , one may write X ∼ N ( μ , σ 2 ) . {\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2}).} Some authors advocate using the precision τ {\textstyle \tau } as

2356-485: Is part of a more general group of parametric kernel methods that employ distributions other than the normal distribution as the kernel elements associated with each point in the set of location data. Recently, the kernel approach to constructing utilization distributions was extended to include a number of nonparametric methods such as the Burgman and Fox's alpha-hull method and Getz and Wilmers local convex hull (LoCoH) method. This latter method has now been extended from

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2432-402: Is the mean or expectation of the distribution (and also its median and mode ), while the parameter σ 2 {\textstyle \sigma ^{2}} is the variance . The standard deviation of the distribution is σ {\textstyle \sigma } (sigma). A random variable with a Gaussian distribution is said to be normally distributed , and

2508-418: 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

2584-868: Is very close to zero, and simplifies formulas in some contexts, such as in the Bayesian inference of variables with multivariate normal distribution . Alternatively, the reciprocal of the standard deviation τ ′ = 1 / σ {\textstyle \tau '=1/\sigma } might be defined as the precision , in which case the expression of the normal distribution becomes f ( x ) = τ ′ 2 π e − ( τ ′ ) 2 ( x − μ ) 2 / 2 . {\displaystyle f(x)={\frac {\tau '}{\sqrt {2\pi }}}e^{-(\tau ')^{2}(x-\mu )^{2}/2}.} According to Stigler, this formulation

2660-1910: The e a x 2 {\textstyle e^{ax^{2}}} family of derivatives may be used to easily construct a rapidly converging Taylor series expansion using recursive entries about any point of known value of the distribution, Φ ( x 0 ) {\textstyle \Phi (x_{0})} : Φ ( x ) = ∑ n = 0 ∞ Φ ( n ) ( x 0 ) n ! ( x − x 0 ) n , {\displaystyle \Phi (x)=\sum _{n=0}^{\infty }{\frac {\Phi ^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}\,,} where: Φ ( 0 ) ( x 0 ) = 1 2 π ∫ − ∞ x 0 e − t 2 / 2 d t Φ ( 1 ) ( x 0 ) = 1 2 π e − x 0 2 / 2 Φ ( n ) ( x 0 ) = − ( x 0 Φ ( n − 1 ) ( x 0 ) + ( n − 2 ) Φ ( n − 2 ) ( x 0 ) ) , n ≥ 2 . {\displaystyle {\begin{aligned}\Phi ^{(0)}(x_{0})&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x_{0}}e^{-t^{2}/2}\,dt\\\Phi ^{(1)}(x_{0})&={\frac {1}{\sqrt {2\pi }}}e^{-x_{0}^{2}/2}\\\Phi ^{(n)}(x_{0})&=-\left(x_{0}\Phi ^{(n-1)}(x_{0})+(n-2)\Phi ^{(n-2)}(x_{0})\right),&n\geq 2\,.\end{aligned}}} An application for

2736-861: The Q {\textstyle Q} -function, all of which are simple transformations of Φ {\textstyle \Phi } , are also used occasionally. The graph of the standard normal cumulative distribution function Φ {\textstyle \Phi } has 2-fold rotational symmetry around the point (0,1/2); that is, Φ ( − x ) = 1 − Φ ( x ) {\textstyle \Phi (-x)=1-\Phi (x)} . Its antiderivative (indefinite integral) can be expressed as follows: ∫ Φ ( x ) d x = x Φ ( x ) + φ ( x ) + C . {\displaystyle \int \Phi (x)\,dx=x\Phi (x)+\varphi (x)+C.} The cumulative distribution function of

2812-632: The Cauchy , Student's t , and logistic distributions). (For other names, see Naming .) The univariate probability distribution is generalized for vectors in the multivariate normal distribution and for matrices in the matrix normal distribution . The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution . This is a special case when μ = 0 {\textstyle \mu =0} and σ 2 = 1 {\textstyle \sigma ^{2}=1} , and it

2888-424: The discrete uniform , Bernoulli , binomial , negative binomial , Poisson and geometric distributions . Important continuous distributions include the continuous uniform , normal , exponential , gamma and beta distributions . In probability theory, there are several notions of convergence for random variables . They are listed below in the order of strength, i.e., any subsequent notion of convergence in

2964-850: The double factorial . An asymptotic expansion of the cumulative distribution function for large x can also be derived using integration by parts. For more, see Error function#Asymptotic expansion . A quick approximation to the standard normal distribution's cumulative distribution function can be found by using a Taylor series approximation: Φ ( x ) ≈ 1 2 + 1 2 π ∑ k = 0 n ( − 1 ) k x ( 2 k + 1 ) 2 k k ! ( 2 k + 1 ) . {\displaystyle \Phi (x)\approx {\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\sum _{k=0}^{n}{\frac {(-1)^{k}x^{(2k+1)}}{2^{k}k!(2k+1)}}\,.} The recursive nature of

3040-699: 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" ( X ( heads ) = 0 {\textstyle X({\text{heads}})=0} ) and to the outcome "tails" the number "1" ( X ( tails ) = 1 {\displaystyle X({\text{tails}})=1} ). Discrete probability theory deals with events that occur in countable sample spaces. Examples: Throwing dice , experiments with decks of cards , random walk , and tossing coins . Classical definition : Initially

3116-406: The integral is still 1. If Z {\textstyle Z} is a standard normal deviate , then X = σ Z + μ {\textstyle X=\sigma Z+\mu } will have a normal distribution with expected value μ {\textstyle \mu } and standard deviation σ {\textstyle \sigma } . This

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3192-886: The weak and the strong law of large numbers 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 . For example, if Y 1 , Y 2 , . . . {\displaystyle Y_{1},Y_{2},...\,} are independent Bernoulli random variables taking values 1 with probability p and 0 with probability 1- p , then E ( Y i ) = p {\displaystyle {\textrm {E}}(Y_{i})=p} for all i , so that Y ¯ n {\displaystyle {\bar {Y}}_{n}} converges to p almost surely . The central limit theorem (CLT) explains

3268-556: The 6 have even numbers and each face has the same probability of appearing. 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 by Ω {\displaystyle \Omega } . It is then assumed that for each element x ∈ Ω {\displaystyle x\in \Omega \,} , an intrinsic "probability" value f ( x ) {\displaystyle f(x)\,}

3344-466: The Taylor series expansion above to minimize computations. Repeat the following process until the difference between the computed Φ ( x n ) {\textstyle \Phi (x_{n})} and the desired Φ {\textstyle \Phi } , which we will call Φ ( desired ) {\textstyle \Phi ({\text{desired}})} ,

3420-459: The Taylor series expansion above to minimize the number of computations. Newton's method is ideal to solve this problem because the first derivative of Φ ( x ) {\textstyle \Phi (x)} , which is an integral of the normal standard distribution, is the normal standard distribution, and is readily available to use in the Newton's method solution. To solve, select

3496-401: The above Taylor series expansion is to use Newton's method to reverse the computation. That is, if we have a value for the cumulative distribution function , Φ ( x ) {\textstyle \Phi (x)} , but do not know the x needed to obtain the Φ ( x ) {\textstyle \Phi (x)} , we can use Newton's method to find x, and use

3572-586: The appendix of a 2017 JMIR article, the home ranges for over 150 different bird species in Manitoba are reported. Normal distribution I ( μ , σ ) = ( 1 / σ 2 0 0 2 / σ 2 ) {\displaystyle {\mathcal {I}}(\mu ,\sigma )={\begin{pmatrix}1/\sigma ^{2}&0\\0&2/\sigma ^{2}\end{pmatrix}}} In probability theory and statistics ,

3648-492: The boundaries of a home range from a set of location data is to construct the smallest possible convex polygon around the data. This approach is referred to as the minimum convex polygon (MCP) method which is still widely employed, but has many drawbacks including often overestimating the size of home ranges. The best known methods for constructing utilization distributions are the so-called bivariate Gaussian or normal distribution kernel density methods . This group of methods

3724-431: The density above is most commonly known as the standard normal, a few authors have used that term to describe other versions of the normal distribution. Carl Friedrich Gauss , for example, once defined the standard normal as φ ( z ) = e − z 2 π , {\displaystyle \varphi (z)={\frac {e^{-z^{2}}}{\sqrt {\pi }}},} which has

3800-410: The derivative gives us the CDF back again, then the random variable X is said to have a probability density function ( PDF ) or simply density f ( x ) = d F ( x ) d x . {\displaystyle f(x)={\frac {dF(x)}{dx}}\,.} For a set E ⊆ R {\displaystyle E\subseteq \mathbb {R} } ,

3876-490: The discrete, continuous, a mix of the two, and more. 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 (or equivalently, the event 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,

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3952-439: The distribution then becomes f ( x ) = τ 2 π e − τ ( x − μ ) 2 / 2 . {\displaystyle f(x)={\sqrt {\frac {\tau }{2\pi }}}e^{-\tau (x-\mu )^{2}/2}.} This choice is claimed to have advantages in numerical computations when σ {\textstyle \sigma }

4028-453: The event E {\displaystyle E\,} is defined as So, the probability of the entire sample space is 1, and the probability of the null event is 0. The function f ( x ) {\displaystyle f(x)\,} mapping a point in the sample space to the "probability" value is called a probability mass function abbreviated as pmf . Continuous probability theory deals with events that occur in

4104-427: 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 any of these events occurs

4180-624: The form of a two dimensional probability density function that represents the probability of finding an animal in a defined area within its home range. The home range of an individual animal is typically constructed from a set of location points that have been collected over a period of time, identifying the position in space of an individual at many points in time. Such data are now collected automatically using collars placed on individuals that transmit through satellites or using mobile cellphone technology and global positioning systems ( GPS ) technology, at regular intervals. The simplest way to draw

4256-401: 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. The law of large numbers (LLN) states that the sample average of

4332-406: The list implies convergence according to all of the preceding notions. 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. Common intuition suggests that if a fair coin is tossed many times, then roughly half of

4408-433: 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

4484-540: 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 . Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The measure theory-based treatment of probability covers

4560-411: 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

4636-412: The parameter defining the width of the distribution, instead of the standard deviation σ {\textstyle \sigma } or the variance σ 2 {\textstyle \sigma ^{2}} . The precision is normally defined as the reciprocal of the variance, 1 / σ 2 {\textstyle 1/\sigma ^{2}} . The formula for

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4712-1207: The probability of a random variable, with normal distribution of mean 0 and variance 1/2 falling in the range [ − x , x ] {\textstyle [-x,x]} . That is: erf ⁡ ( x ) = 1 π ∫ − x x e − t 2 d t = 2 π ∫ 0 x e − t 2 d t . {\displaystyle \operatorname {erf} (x)={\frac {1}{\sqrt {\pi }}}\int _{-x}^{x}e^{-t^{2}}\,dt={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt\,.} These integrals cannot be expressed in terms of elementary functions, and are often said to be special functions . However, many numerical approximations are known; see below for more. The two functions are closely related, namely Φ ( x ) = 1 2 [ 1 + erf ⁡ ( x 2 ) ] . {\displaystyle \Phi (x)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right]\,.} For

4788-470: 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 3 6 = 1 2 {\displaystyle {\tfrac {3}{6}}={\tfrac {1}{2}}} , since 3 faces out of

4864-733: The probability of the random variable X being in E {\displaystyle E\,} 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 (including discrete random variables) that take values in R . {\displaystyle \mathbb {R} \,.} These concepts can be generalized for multidimensional cases on R n {\displaystyle \mathbb {R} ^{n}} and other continuous sample spaces. The utility of

4940-450: The probability that X will be less than or equal to x . The CDF necessarily satisfies the following properties. The random variable X {\displaystyle X} is said to have a continuous probability distribution if the corresponding CDF F {\displaystyle F} is continuous. If F {\displaystyle F\,} is absolutely continuous , i.e., its derivative exists and integrating

5016-560: The sequence of random variables converges in distribution to a standard normal random variable. For some classes of random variables, the classic central limit theorem works rather fast, as illustrated in the Berry–Esseen theorem . For example, the distributions with finite first, second, and third moment from the exponential family ; on the other hand, for some random variables of the heavy tail and fat tail variety, it works very slowly or may not work at all: in such cases one may use

5092-581: The standard normal cumulative distribution function, Q ( x ) = 1 − Φ ( x ) {\textstyle Q(x)=1-\Phi (x)} , is often called the Q-function , especially in engineering texts. It gives the probability that the value of a standard normal random variable X {\textstyle X} will exceed x {\textstyle x} : P ( X > x ) {\textstyle P(X>x)} . Other definitions of

5168-783: The standard normal distribution can be expanded by Integration by parts into a series: Φ ( x ) = 1 2 + 1 2 π ⋅ e − x 2 / 2 [ x + x 3 3 + x 5 3 ⋅ 5 + ⋯ + x 2 n + 1 ( 2 n + 1 ) ! ! + ⋯ ] . {\displaystyle \Phi (x)={\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\cdot e^{-x^{2}/2}\left[x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{3\cdot 5}}+\cdots +{\frac {x^{2n+1}}{(2n+1)!!}}+\cdots \right]\,.} where ! ! {\textstyle !!} denotes

5244-600: The standard normal distribution, usually denoted with the capital Greek letter Φ {\textstyle \Phi } , is the integral Φ ( x ) = 1 2 π ∫ − ∞ x e − t 2 / 2 d t . {\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt\,.} The related error function erf ⁡ ( x ) {\textstyle \operatorname {erf} (x)} gives

5320-520: The standard normal distribution. This variate is also called the standardized form of X {\textstyle X} . The probability density of the standard Gaussian distribution (standard normal distribution, with zero mean and unit variance) is often denoted with the Greek letter ϕ {\textstyle \phi } ( phi ). The alternative form of the Greek letter phi, φ {\textstyle \varphi } ,

5396-400: The subset {1,3,5} is an element of the power set of the sample space of dice 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. Probability is a way of assigning every "event" a value between zero and one, with the requirement that

5472-558: 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

5548-407: 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 theory 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

5624-756: The ubiquitous occurrence of the normal distribution in nature, and this theorem, according to David Williams, "is one of the great results of mathematics." 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 X 1 , X 2 , … {\displaystyle X_{1},X_{2},\dots \,} be independent random variables with mean μ {\displaystyle \mu } and variance σ 2 > 0. {\displaystyle \sigma ^{2}>0.\,} Then

5700-561: The σ-algebra F {\displaystyle {\mathcal {F}}\,} is defined as where the integration is with respect to the measure μ F {\displaystyle \mu _{F}\,} induced by F . {\displaystyle F\,.} Along with providing better understanding and unification of discrete and continuous probabilities, measure-theoretic treatment also allows us to work on probabilities outside R n {\displaystyle \mathbb {R} ^{n}} , as in

5776-403: Was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics . 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 (for example the " problem of points "). Christiaan Huygens published

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